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STUMNTS TEXT
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INTRODUCTIO;; TO
MATRIX ALGEBRA
SCHOOL MATHE#ATlCS STUDY
YALE UNIWRSCTY ?RESS
GROUP
School Mathematics Study Group
Introduction to Matrix Algebra
Ii~troductionto Matrix Algebra
Strrdetlt's
Text
Prcparcd u n h r th ~est~pen.kjon
of
rhc Panel on Sample T e x r h k s
of rhc School Mathcmarm Srudy Group:
Frank
B. Allen
Edwirl C. Douglas
Donakl E. Richmond
Cllarlm E,Rickart
Hcnry Swain
Robert J. Walkcr
Lyonr Township High
School
Tafi S C h I
Williamr Collcgc
Yale Univcrriry
New Trier Township High S c h d
Cornell University
Ncw Havcn a d h d o n , Y a1c University Press
Coyyrig!lt O I*,
r p 5 r by Yak Ut~jversiry.
Printed in rhc Ur~itcdStarm af America.
A]] r i ~ h t srescwcd. Ths h k may not
ltrc repcduccd, in whdc or in part, in
any form, wirho~~t
written permission from
the publishen.
FinanciaZ slspporr for the S c b l Mathemaria
Study Group has km providcd by rhe Narional
Scicncu: Foundation.
The i n c r e a s i n g contribution of mathematics to the culture of the d c r n
w r l d , as well as i t s importance as a v i t a l part of s c i e n t i f i c and humanistic
tdueation, has mndc i t essential that the m a t h e ~ u t i e ai n our echool@ be bath
wZ1 selected and e l l taught.
t j i t h t h i s i n mind, the various mathc~atical organLtaeions i n the United
Seatss cooperated i n the forumtion of the School F ~ t k m a e i c sStudy Group (SPSC).
SMG includes college and unirersiley mathmaticlans, eencbera of m a t h w t i e s a t
a l l lavels, experts i n education, and reprcscntarives of science and tcchrmlogy.
Pha general objective o f SMG i a the i m p r o v m n t a f thc? teaching of =themtics
in the sehmla of t h i s country. The l a t i o n u l Science Foundation has provided
subs tantiaL funds for the support of this endeavor.
One of the pterequlsitea for the improvement of t h e teaching of math&matira
in our ~ c h m t si e on improved c u r r i e u ~ ~ n
which
s
takea account of thc incrcasing use of oathematice Ln science and teehnoLogy end i n other areas o f knowledge
and a t the a m timc one v f i i ~ href l o e t a recent advancta i n mathem~tica i t s c l f .
One of thc f l r n t projects undartaken by SrSC was t o enlist a group of outstanding
mathematiclans and nrarhematica teachern t o prepare a nariea of textbooks uhich
would illustrmre such A n improved curriculum.
The professional arathcmatLcLanlr in SMSG belleve that the mathematics p r P
eented In this t t x r is vaLuahLc fat a l l w2 l-ducatcd
citlzene i n our society
t o know 6nd t h a t it ia tqmrtant f o r the prceollage atudent: to learn i n preparation far advanced vork i n thc f i e l d . A t thc time t i n e , teachers in SWC believe
t h a t it is presented in rrueh a farm that i t can ha r ~ a d F l ygrasped by students.
!
!
In most inntanees the mnterlal will have a inmiltar n o t e , he the prcsentation and the point of vicv vill be diftcrsnt. Samc matcrial w i l l be e n t i r e l y
neu t o ehc traditional curriculum, This i s ar i t should be, for ~ A a t h c m & t i ~i &
a
a ltving and an cvcwrawinff subject, and Rat a dead and frozen product of ant i q u i t y . This heakthy fusion o f the old and Ehc ncu should lead students to a
better undcsstsnding of the basic concepts and structure of mthemtics and
provide a f i m r foundation for understanding and use of ~ t h c m a t i c sin a
gcitntlfic sociery.
It la n o t intendad that this book be regarded an the mly d c f i n i t l v a way
of presenting good mathcmatica t o student# a t t h i s Level. Instead, I t should
be thought of as e ample of the kind of i q r w e d currieulua that we need and
as a source of nuggestiana for the authora of c-rclal
textbka.
It is
sincerely hoped that these t e x t s wLl1 lead the way coward inspiring a more
meaningful reaching o f Hathemnckea, t h e Wcca and Servant of tho Sctencea.
5.
(b)
Name the entries in the 3rd row.
(c)
Name the entries in the 3rd column.
(d)
Name the entry
(e)
Forwhatvalues
i, j
is
b
(f)
For what values
i, j
is
b
(g)
W r i t e the transpose
(a)
Write a
3 x 3
matrix a l l of whose entries are whole numbers.
(b)
Write a
3
4
matrix none of whose entries are whole numbers.
(c)
Write a
5 x 5
matrix having a l l entries in i t s f i r s t two rows
X
b12.
ij
ij
+0?
= O?
B ~ ,
p o s i t i v e , and a l l entries in its l a s t three rows negative.
6.
1-3.
(a)
How many entries are there in a
(b)
In a
(c)
Xn en
(d)
In an rn x n matrix?
4 x 3
n x n
2 x 2
matrix?
matrix?
matrix?
Equality of Matrices
Two matrices are equal provided they are
of
the same order and each entry
For example,
in the f i r s t is equal t o the corresponding entry Ln the second.
but
Definition 1-2.
Two matrices
A
and
B
are equal,
A =
B,
if and only
if they are of the same order and their corresponding entries are equal.
Thus,
36
ordinary algebra of numbers insofar as multiplication is concerned?
L e t us consider an example that will y i e l d an answer t o the foregoing
question.
Let
If we compute
AB,
we find
Now, if we reverse the order of the factors and compute
AB
Thus
BA,
we find
BA are d i f f e r e n t matrices!
and
For another example, let
I
Then
while
Again
AB and BA are d i f f e r e n t matrices; they are not even of the same order!
Thus we have a first difference between matrix algebra and ordinary
algebra, and a very significant difference it is indeed.
men
we
numbers, we can rearrange factors s i n c e the commutative law h o l d s :
x
E
R
and
y
E
R,
we have
xy = yx.
multiply real
For all
When we multiply matrices, w have no
37
such law and w e must consequently b e careful t o take the factors i n the order
given.
W
e must consequently d i s t i n g u f s h between the reaul t o f multiplying
on the right by
by
A
to g e t
A
to g e t
and the r e s u l t of multiplying
BA,
B
B
on the left:
In the algebra of numbers, these two operations of "right
AB.
m u l t i p l i c a t i o n t 1 and "left muLtiplFcation" are the same; in matrix algebra, they
are n o t necessarily the same.
Let u s explore some mote differences!
A =
Patently,
[; ;]
# 0 and B # g.
A
thus, we f i n d
and
8 =
Let
[-;_93]
But i f we compute
AB,
.
we obtain
. Again, l e t
d
Then
The second major difference between ordinary algebra and matrix algebra is
that the product of tw, matrices can be a zero matrix without either factor
being a zero matrix.
The breakdown f o r matrix algebra o f the law that
that
xy = 0
only i f either
x
or
y
xy = y x
is zero causes additional difference^.
For instance, f o r real numbers ue know that if ab = a c ,
then
b =
c.
Proof,
and
a
+ 0,
T h i s property is called the cancellation law for multiplication.
We d i v i d e d the proof into afmple atepB:
(b)
ab ac,
ab - a t = 0,
(c)
a(b
{a)
and of the law
- c)
= 0,
For tmtricer , the above s t t p from (el t a Cd) f a i l d and rhs proof i e not
v a l i d . In fact, AB can be aqua1 t o AC, with A 3 2, y e t B
C g Thud, l e t
+
Then
but
Let us consider another d i f faranee.
have a t moer
equation
Praof
mt
.
t
v
Wi h o w t h t r real
nmber
a
can
that is, thars arc a t m o s t tw rocrre of the
square
~
root.;
a.
Again, ws give the r i q l h s t s p l o f the proof:
Supp6c that
py
-
them
a;
W,
XX
xx-yy-0,
(X
Yx
-
-
Y)(X
+ Y ) ' = + (v+ x y )
KY.
Prm (dl and
la),
Prom (cl and ( I ) ,
Therefore, sither
- y)(x
(x
(x
+ y)
- y)(x + y)
- yy,
-+
- --
x
-y
Therefore, either x
y
0
or
or x
m0.
n.
y
0.
x
m
y.
For matrices, statement (c) is f a l s e , and therefore the steps t o ( f )
(g) arc i n v a l i d .
Even i f ($1 vtre valid, the #tap from (g) t o (h) f a i l & .
ISM. 1-81
and
39
Thsrefore, the forcgofng proof i e invalid if ue t r y t o a p p l y i t t o m t r i c a a .
fact, k t is false that a matrix can have a t =st t w o square roots: b k have
In
Ttluo t h o matrix
has thc four d i f f e r a n t square roots
There are mra!
By giving
x
Given any number
x
#
0,
we
have
ady one of an i n f i n i t y of different real values, w obtain an
i n f i n i t y o f d i f f e r e n t square roots a f the matrix 1:
Thub
roots!
the very stnple
You can see,
most t w o squaro
2 x 2 matrix
I has i n f i n f t e l y many d i s t i n c t square
then, that the fact that a real or c q l c x nunbct has a t
m a t e Lsr by no means trivial.
2.
k k c the cakculations of EKrrcts-a L for tha artrictr
3.
k
t
and
A
Calculate
B be as in Excrciae
AI,
2, and let
IB, and
IA, RX,
(AX)B.
4. Ltt
Show by c m p u t a t i o a t h a t
where
(A)
(A
+ R I M + BS
+ A'
(b)
(A
+ B)(A-
+ -
A
and
02
B)
denole
+~
M
A B+ 8'.
and
BB, rempectlvtly.
6.
Find a t Least t3 squara roots o f the m t r i x
7.
Show that: the m t r i x
a a t i s f l c s the aquatian A'
-
2.
How many 2 x 2 matrices .atla
fying
thia
equation can you flad?
8.
Show t h a t the matrix
s a t i a f i e r the cquarion k3
1-9.
0.
Properties of Hatrix Hultiplication (Concluded)
tb have seen that tvo basic laus governing multiplication i n the algebra
of ordinary ambers break dowa when k t comas t o matrices.
and the cancellation law do n o t hold.
t h e c-tativc
law
A t t h i ~point, you night fear a total
all thc other f m i l i a r law.
This Is not the c a m . Aside from the
tw laws aentloned, and the f a c t that, as wc ehall rec later, many matrices do
not have aulsiplicativc invarues (reciprocals), thc other bani c Paw o f
ardinary algebrn generally r-in
v a l i d for mutriccs. The assoedative Law
holda for the w l t i p l i e a t i m of mtricea and thare are dtatributive levr that
u n i t e addition and multiplication.
A feu e x m p l r s will aid us In understanding the lam.
collapas
of
kt
[see.
1-81
42
Then
and
Thus, in this case,
and
so that also
Since multiplication is not c m r a t i v e , we cannot conclude from Equation
(2)
that the distrtbutive principle is v a l i d with the factor
hand s i d e of
B
+ C.
A
on the right-
Having illustrated the Left-hand distributive law, w
43
now i l l u s t r a t e the right-hand distributive law with the following example:
We have
and
Thus,
(I3
+ C)A
= BA
+ CA.
You might note, in passing, t h a t , i n the above example,
These properttes o f matrix mu1tiplication can be expressed as theorems,
as f o l l o w s .
Theorem 1-55.
If
then
(AB)C
Proof. (Optional .) We have
r:
A(BC)
.
Since the o r d e r of a d d i t i o n is arbitrary, we know that
Hence,
Theorem 1-6.
then
A ( B f C) = AB
Proof.
If
+ BC.
(Optional.)
(8
We have
+ C)
=
[bjk
+
'jk]
p x n'
aij ( b j k + c . )
J~
1
mxn
] ...
-
AB
f AC.
Hence,
Theorem 1-7.
If
=
then
(B
+ CIA
[bjk] p x n *
= BA
=
['jk]
p ~ n 'and
A =
[%i]
nxq
+ CA.
Proof.
The p r o o f is similar t o that o f Theorem 1-6 and will be l e f t as
an exercise for the s t u d e n t .
It should be noted t h a t if the
comtative
Law h e l d for matrices, it would
be unnecessary to prove Theorems 1-6 and 1-7 separately, since the two stare-
men ts
would be equivalent.
For matrices, however, the two statements are n o t e q u i v e
l e n t , even though borh are true.
statements like
and
can be false for matrices.
The order of factors is most important, since
46
m x n
Earlier we defined the zero matrix o f order
and showed that it: i s
the identiry element for matrix addition:
where A i s any matrix of order m x n. This zero matrix plays the same role
in t h e m l t i p l f c a t i o n of =trices as the number zero does in the ~mltiplicatition
of real numbers. For exrtm~le,we have
Theorem 1 4 .
For any matrix
we have
Omxp
The proof is
A p x n = 0m x n and A p x n 0n x q = 0p x q '
easy and is l e f t t o the student.
Now we may be wondering If there is an i d e n t i v element for the multiplica-
tion of matrices, namely a matrix that p l a y s the same r o l e a e the number 1 does
in the multiplication of real numbers.
There
(Far all real numbers
a, l a = a = al.)
such a matrix, c a l l e d the unit matrix, or the identity matrix for
multiplicatfon, and denoted by the symbol I.
The matrix
12, namely,
is called the u n i t matrix of order 2 .
The matrix
is called the unit matrix o f order 3 .
In general, the unit marrix o f order
n
i s the aquare matrix
[eij] nx n
such that
e
ij
1
forall
i - j
and
e
ij
forall
= Q
#
j (
i
2
.
n
; =
1
,
.n
W e n o w s t a t e the
important property o f the unit matrix as a theorem.
Theorem 1-9.
Proof.
product
all
Am
k
j
If A
is an m x n
matrix, then
By d e f i n i t i o n , t h e e n t r y in the
i s the sum
i-th
AIn = A
row and
j-th
and
ImA
A.
column of the
+ a e
+ . . . + a e .
Since e
ilelj
12 2j
i n nj'
kj = 0
a l l terms but one in this Bum are zero and drop o u t . We are
a
for
Thus the entry in the i-th row and j-th column of the
a .e = a
$J j j
ij'
product is the same as the corresponding entry in A. Hence A l n = A . The
leftwith
equality
Imh = A
may be proved the same way.
In most situations, it: is
not
necessary t o specify the order of the unit matrix since the order is inferred
from the context.
Thus, for
ImA = A
AI,,
we write
IA * A = AX.
For example, we have
and
Exercises 1-9
43
Test the formulas
+ C)
A(B
+ AC,
= AB
(B 4- CIA = BA
+ CA,
A(B 4- C)
AB
+ CA,
= BA
+ CA.
-
+ C)
A(O
Which are correct, and which are false?
2.
Let
A Show that
3.
2,
AB
001
[;
but
BA =
Show t h a t f o r all matrices
A
[;
and
;I.
.==
2.
of the form
and
B
AB =
BA.
we have
Illustrate by assigning numerical values to a,
a, b, c, and d i n t e g e r s .
4.
Find the value of x
5.
For the matrices
show rhat
6.
b,
and
c,
for which the following product is
d,
1:
0
0
0
0
0
0
0
0
-
0
1
0
0
1 2 0
=
CA,
AB
1
BA,
rhat
AC
and that
Show t h a t the matrix
fsec. 1-93
BC
1
CB.
0
0
with
= I.
satisfies the equation
Find at least one more s o l u r i o n of t h i s
equation.
7.
Show t h a t for all matrices
A
of the form
we have
Illustrare by assigning numerical values to
8.
a
and
b.
Let
Compote the following:
If
AB =
- BA,
A
ED,
(el
m,
( f
FE.
B are s a i d
and
c l u s i o n s can be drawn concerning
'I. Show that the m a t r i x A =
A'
(a>
- 5A + 71 = 2.
D,
[-: :]
.-
t o be
bhat: con-
and
E,
F?
is a solution o f the equation
10. Explain why, in matrix algebra,
(h 4- 8) ( A - B )
except in special cases.
# A
2
- B
2
Can you devise two m a t r i c e s
A
and B
will i l l u s t r a t e the inequality? Can you d e v i s e two matrices
A
that
and
B
50
that will illustrate the special case?
(Hint:
Use square metrices of
order 2.)
*
11.
Show that if V and W
12,
Prove that
are n x 1
colum vectors, then
( A B ) ~= B ~ A ~asrruming
,
that
A
and B are c c m f o l l e for
multiplication,
no tation, prove the right-hand d i s t r i b u t i v e law (Theorem 1.7)
13. Using
Sumwry
1-10.
In this introductory chapter
ve have d e f i n e d several operations on
matricea, such as a d d i t i o n and multiplication.
Theee operatione d i f f e r from
those of elementary algebra i n tbat they cannot always be performed.
we
.
do not add a
matrix and a
4 x 3
nor
2 x 2
3 x 4
3 x 4.
matrix t o a
3 x 3 matrix; again, though a
Thus,
4x 3
matrix can be multiplied together, the product i e neither
More importantly, the c-tative
law for multiplication
do not hold.
There La a third significant difference that we shall explore more f u l l y
and the cancellation law
in later chapeere but shall tntroduce now. Recall that the operation of
subtraction was closely associated with that o f addition. In order t o solve
equations of the form
it fsconveniept t o employ the additive inveree, or negative,
4.Thus, if
the foregoing equation holds, then we have
A+x+(-A)
X + A + (-A)
x+g=
-
B+(-A),
= B+(-A),
B-A*
X-I-A.
b you know, every matrix A
e negative -A*
Now "division" i s closely
9
associated w i t h m l t i p l i c a t i o n in a parallel manner.
In order to solve
equation8 o f the form
multiplicative inverse (or reciprocal), which is
we would analogously employ
denoted by the symbol A-'.
The defining property is A-'
A = I = M - ~ . This
enables us to solve equations of the form
Thus i f the foregoing equation holds, and i f A has a mu1 tiplicarive Lnverse
-1
A , then
Now, many matrices other than the zero matrix
2
do not possess multiplicative
inverses; for i n s mnce,
are matrieee of this sort.
This fact constimtes a very significant difference
between the algebra o f -trices
and the algebra of real numberr.
In the next
tvo chapters, we shall explore the problem of matrix inversion in depth.
Before closing this chapter, w e should note that matrices arising in
s c i e n t i f i c and industrial applications are narch larger and their entries much
more complicated than has been the case in this chapter.
As you can imagine,
the computations involved d e n dealing with larger matrices (of order 10 or
more), which is usual in applied work, are ao extensive a8 to discourage their
uea i n hand computations.
Fortunately, the recent development of higltapeed
electronic computers has largely overcome this difficulty and thereby has made
it mre feasible t o apply matrix methods i n many areas of endeavor.
Chapter 2
THE ALGEBRA OF
2-1
.
2
X
2 MTRICES
Introduction
In Chapter 1,
we considered the elementary operations of addition and
This algebra is similar in many
respecta to the algebra of real numbers, a 1 though there are important differ-ences. Specifically, we noted that the connnutative law and the cancellation
multiplication for rectangular matrices.
law do not hold f o r matrix multiplication, and that division is not always
possible.
With matrices, the whole problem of d i v i s i o n i s a very complex one; i t i s
centered around the existence of a multiplicative inverse.
question:
Let us ask a
If you were given t h e matrix equation
could you solve i t for the unknom
4 x 4 matrix
X?
Do n o t be dismayed
Eventually, we s h a l l learn methods of solving t h i s
if your answer is "No."
equation, but the problem is complex and Lengthy.
In order to understand
this problem in depth and at the same time comprehend the f u l l significance
of the algebra we have developed so far, we shall largely confine our attention
i n t h i s chapter t o
a special subset of the set of all rectangular matrices;
2 x 2
namely, we shall consider the a e r of
square matrices.
When one s tsnds back and takes a broad view of the many d i f f e r e n t kinds of
numbers that have been studied, one sees recurring patterns.
us look at the rational numbers for a moment.
can add and m l t i p l y .
granted.
This
statement
But i t is not true of a l l
For Lnstance, let
Here is a set of numbers that
WE
is so simple that we almost rake it for
s e t s , so let:
us g i v e a name to the n o t i o n
that: is involved.
Definition 2 4 .
a f i r s t member
(i)
a
of
A set
S
S
i s said to be closed under an operation
and a second member b
the operation can be performed on each
of
a
S
if
and b
of
S,
R on
5rrf a r each
(ii)
member of
a
and
b
of
the r e s u l t of the operation i e a
S,
S.
For example, the set o f p o s i t i v e i n t e g e r s i s not closed under the operation
of division since for some p o s i t i v e integers
a
and
b
the ratio a/b
i s not
a positive integer; neither is the s e t of rational nmbers closed under division,
since the operation cannot be perfomed if
b = 0 ; but the set of positive
rational numbers is closed under d i v i s i o n since the quotient o f two positive
rational numbers is a positive rational number.
Under addition and multiplication, the s e t of rational numbers s a t i s f i e s
the following postulates:
The s e t is closed under addition.
Addition i s c o m t a t i v e .
Addition i s a s a o c i a r i v e .
There i s an i d e n t i t y member ( 0 ) for addition.
There is an additive inverse member
-a
for each member
a.
The set i s closed under multiplication.
Mu1 t i p l i c a t i o n is cwnattative
.
Efultiplication i s a s s o c i a t i v e .
There is an i d e n t i t y member (1) for multiplication.
-1
There i s a multiplicative inverse member a
other than 0 .
for each member
a,
. . .
Multiplication is d i s t r i b u t i v e over a d d i t i o n .
Since there exists a rational multiplicative inverse for each rational number
except
0,
division (except by 0) is always possible in the algebra of
rational numbers.
where
a
and
b
In other words, all equations of the form
are rational numbers and
the algebra of rational numbers.
a # 0, can be solved for
For example:
x
in
I n order t o solve the equation
we multiply both sidea of the equation by
of - 2 1 3 .
- 3/2,
the m u l t i p l i c a t i v e inverse
Thus we obtain
which i s a rational number.
The foregoing set of postulates is satisfied also by the s e t of real
numbers.
Any s e t that satisfies such a e e t of postulates i s called a f i e l d .
Both the s e t o f real numbers and the
s e t of r a t i o n a l s , which i s a subset
the s e t of real numbers, are fields under addition and m u l t i p l i c a t i o n .
are many systems t h a t have this same pattern.
of
There
In each of these systems,
division (except by 0) is always possible.
Now our imediate concern is to explore the problem of division in the set
of matrices.
There is no blanker answer that can readily be reached, although
there i s an answer that we can f i n d by proceeding stepwise.
limit our discussion to the s e t of
2 x 2
matrices.
A t f i r a t , l e t us
We do t h i s n o t o n l y to
consider division in a smaller domain, but a l s o to study i n d e t a i l the algebra
associated w i t h t h i s subset.
A more general problem o f . m t r i x d i v i s i o n will be
considered in Chapter 3 .
Exercises 2-1
1.
Determine which of the following sets are closed under the s t a t e d
operation:
(a)
the s e t of integers under addition,
(b)
the s e t of even numbere under multiplication,
(c)
the s e t 111 under multiplicarion,
(d)
the s e t of positive irrational numbers under division,
[ssc. 2-11
56
2.
(e)
the s e t o f integers under the operation of squaring,
(f)
the s e t of numbers A = Ex: x > 3)
under addition.
Detennine which of the following statements are true, and s t a t e which of
the indicated operations are c o m u t a t t v e :
(a)
2-3=3-2,
(b)
4+2=2+4,
(c)
3+2=2+3,
(d)
x
a
(e)
a
(f)
Pq = q p ,
-b
Jb
=
= b
- a,
+ Ja,
a
a
and
and
b
positive,
real,
b
real,
P and q
F
le 2 = 2
(g)
3.
+
+ fi.
Determine which of the following operations 3, defined for p o s i t i v e
integers in terns of a d d i t i o n and mu1tlplication, are c m u t a t i v e :
(forexample,
(a) x 3 y P x + 2 y
4.
(e)
x * y = x Y,
(f)
x * y = x + y + l .
Determine which of the following operations
233=2+6=8),
*,
d e f i n e d for p o s i t i v e
integers i n terms o f addition and mu1 tipLfcation, are associative:
5.
(a)
x * y = x + Z y
(d)
x
(el
x
(f)
x
*
Y
*Y
*y
(for enample, ( 2
*
3)
*
4 = 8
*
4
a
161,
XI
=
J G s
= xy
+ 1.
Determine whether the operation
*
9, that is, determine whether
*
x
is distributive over the operation
(y 5
z) = ( x
* y) + ( x *
2)
and
*
( y f z)
x = (y
* x)
%
( Z
*
X)
,
fi.
*
and
where the operations
are
d e f i n e d For positive integers i n terms o f addition and mu1 tiplicetion of
real numbers:
x +
(a)
x + y =
(b)
x3yE2x+2y,
(c)
x * y =
Why is the answer
x +
x*yEllIY;
y,
x
y+1,
*
y
=
1
xy;
x * y - x y .
the same I n each case for Left-hand d i s t r i b u t i o n as it
is for right-hand d i s rribution?
6.
In each of the following examples, determine if the specified s e t , under
a d d i t i o n and multiplication, constitutes a f i e l d :
(a)
the s e t of all positive numbers,
(b)
t h e set: of all rational numbers,
(c)
the set of ell reel numbers of the form
a and b are integers,
+ bfi,
the e e t of all complex numbers o f the form a
and b are real numbers and i =
(d)
a
2-2.
a
The Ring o f
2 x 2
Matrices
Since we are confining our attention t o the subset of
it i s very convenient t o have a symbol for this subset,
s e t of all
2 x 2
matrices.
If
A
f
where
bi,
2 x 2
tle let
where
matrices,
M
denote the
is a member, or element, of t h f s s e t , we
express t h i s membership symbolically by
A E M.
S i n c e all elements of
M
are
matrices, our g e n e r a l definitions of addition and multiplication hold for t h i s
subset.
The s e t
M
is not a f i e l d , as defined in Section 2-2,
since M
does n o t
have a l l the properties of a f i e l d ; for example, you saw in Chapter 1 t h a t
multiplication is not commutative in
we have
M.
Thue, for
Let us now consider a less r e s t r i c t i v e sort of mathematical system known
as a ring;
- t h i s name is
Definition 2-2.
usually aktribured t o David Hilbert (1862-1943).
A r i n g is a s e t with two operations, called addition and
mu1tiplication, that possesses the following properties under a d d i t i o n and
multiplication:
The s e t is closed under a d d i t i o n .
Addition is c o m t a t i v e
.
Addition i s associative
.
There is an i d e n t i t y element for addition.
There is an additive inverse for each element.
The s e t is closed under mu1 tiplieation.
Multiplication is associative.
Multiplication is distributive over addition.
Does the s e t
M
s a t i s f y these properties?
Consider the s e t of a l l real numbers.
b u t the answer i a not q u i t e obvious.
'
It seems clear t h a t i t does,
This s e t is a field because there e x i e t s , among other things, an additlve
inverse for each number in thfs set.
o f the real numbers.
Now the p o s i t i v e integers are a subset
Does thLs s u b s e t contain an a d d i t i v e inverse for each
element? Since we do not have negative integers i n the s e t under consideration,
the answr i e "No"; therefore, the s e t of positive integers irr not a f i e l d .
Thus a subset does not: necessarily have the same propertlea aa the complete
set.
To b e certain that the s e t
M
is a ring, ue must systematically make sure
that each ring criterion is s a t i s f i e d .
For the m a s t part, our proof w i l l be a
reiteration of the material i n Chapter 1, s i n c e the general properties o f
matrices w i l l be valid for the subset
2 x 2
matrices i s a
2 x 2
M
of
2 x 2
matrices.
matrix; that i s , the set
For ewmple,
[aec. 2-21
i6
The sum of twu
c l o s e d under a d d i t i o n .
fie general proofs of cormartativity and associativity are valid.
The unit
matrix is
the zero matrix i s
and the additive inverse of the matrix
When we consider the multiplication of
2 x 2
matrices, we must first v e r i f y
that the product is an element of this s e t , namely a
2 x 2
matrix.
&call
that the number of rows in the product is equal t o the number of rows in the
lef t d n d factor, and the number o f coluarms i a equal t o the number of columns
in the right-hand factor. Thus, the product of two elements of the s e t M
must be an element of this s e t , namely a
is c l o s e d under mu1tiplication.
2 x 2
matrix; accordingly, the s e t
For example,
The general proof of a s s o c i a t i v i t y i s valid for elements of
for rectangular matrices.
of M
M,
s i n c e it , i s v a l i d
Also, both of the distributive laws hold for elements
by the same reasoning.
For example, t o illustrate the ascrociative law
for multiplication, ue have
[see. 2-21
and also
and t o illustrate the left-hand distributive Law, we have
and also
Since we have checked that each of the ring postulates i s fulfilled, we
have shown that the s e t
multiplication.
Theorem
M of
2 x 2
matrices is a ring under addition and
We s t a t e this result fonaally as a theorem.
2-1.
The set M
2 x 2
of
matrices i s a ring under addition
and trmltiplication.
Since the list o f defining properties for a f i e l d contains a l l the d e f i n i n g
propertier for a ring, i t follows that every f i e l d is a r i n g .
statement i a n o t true; for example, we now know that the s e t
matrices i s a ring but not a field.
that is, for each
U =
Thus the s e t
M is
a ring
M
of
2 x 2
The set H bas one more of the f i e l d
properties, namely there is an i d e n t i t y element
for multiplication in M;
But the converse
A
-
A
€
M we have
AX.
with an identity element.
[see.
2-21
61
A t this time, we should verify that the comnutative law for multiplication
and the cancellation law are not valid in H by giving counterexamples.
Thus
we have
but
so that the commutative Law for multiplication does n o t hold.
Also,
so that the cancellation law does n o t hold;
Exercises 2-2
1.
Determine i f the set of a l l integers
18
a ring under rhe operations o f
addition and multiplication.
2.
Determine which of the fallowing s e t s are rings under a d d i t i o n and
mu1 tiplication:
3.
the form a
+b
A,
where
(a)
the s e t of numbere
are integers ;
(b)
the s e t of f o u r f o u r t h roots of unity, namely,
and -i;
(c)
the s e t of numbers
of
a/2,
where
a
e
and
+L, -1,
b
i,
is an integer.
Determine If the set o f a11 u t r t c e ~o f the fom
[t :],
with
a e R,
forms a ring under addition and multiplication as defined for matricea.
4.
Determine i f the s e t of a l l matricea of the form
forms a ring under addition and multiplication a@ defined for matrices.
62
The Uniqueness of the &ltiplicative
2-3.
Inverse
Once again we turn our attention to the problem o f matrix d i v i s i o n .
As we
have seen, rhts problem arises when we seek to solve a matrfx equation of the
form
L e t us look a t a parallel equation concerning real numbers,
Each nonzero number a has a reciprocal l l a ,
B' .
which is often designated
Since multiplication of real numbers i s
-1
commutative, it follows that a a = 1. Hence if a l a a nonzero number,
then there is a number b, called the multiplicative inverse of a, such that
e a l = 1.
Its defining property i s
-1
ab, = 1 = ba
Given an equation
us
to
ax =
C,
where
f i n d a solution for x;
a f 0,
(b = a
the multiplicative inverse
b
1.
enables
thus,
b(ax)
= bc,
( b a h = bc,
lx = bc,
X
= h.
Now our question concerning division by matrices can be put in another way.
A B
M,
i s there a
issatisfied?
verse,
80
B
E
M for which the equation
-L
W e s h a l l e m p l o y themoreauggeativenotation A
that our question can be restated:
which the equation
If
for t h e i n -
Is there an element A - ~ E M
for
63
i s satisfied?
s t a t e it
Since we shall often be using t h i s d e f i n i n g properw, let us
formally as a definition.
Definition 2-3.
A
If
A E
M,
then an element
A-'
of
M
is an inverse of
provided
If there were an element
B corresponding
to
each element
A
f
M such
tht
AB = 1 s BA,
then we could solve a l l equations of the form
AX = C,
since we m u l d have
B(AX) = BC,
(BA)X =
BC,
TX
BC,
-
X = BC,
and clearly t h i s value s a t i s f i e s the original equation.
From the fact that there is a multiplicative inverse for every real number
except zero, we might: wrongly infer a parallel conclusion for matrices.
stated in Chapter 1, not a1 1 matrices
As
have inverses. h r knowledge that 0
has no inverse suggests that the zero matrix 0 has no inverse. This is true,
since we have
-ox
for all
X c
M, so that there
=
0
cannot be any
-ox = I.
[set. 2-31
X
E
PI such that
92
Galois (1811-1832),
Group Theory.
to whom is c r e d i t e d the origin of the systematic study of
Unfortunately, Galois was killed in a duel at the age of 2 1 ,
i m e d i a t e l y a f t e r recording some of his most notable theorems.
Exercises 2-6
1.
Determine whether the following s e t s are groups under mulkiplication:
(b)
'I,
-I,
K,
-K,
where
2.
Show that the s e t a f a l l elements of
M
o f the form
c o n s t i t u t e s a group under multiplication.
3.
Show that the set of all elements of
[: :]
constitutes a
,
where
t E
R.
s
e
M
R,
of the form
and
group under multiplicarion.
show t h a t the s e t
(A, A ~ A
, ~ )
[see. 2-61
t
2
- s2 = 1,
93
Plot the corresponding points i n the
i s a group under multiplication.
plane.
5.
Let
Show that the a e t
1s this true i f
is a group under multiplication.
T
is
invertible
matrix?
6.
Show that the set: o f a l l elements of the form
[: :],
with
a
7.
a
and
b
R,
b E R,
and
ab
-
1,
If you p l o t all of the p o i n t s
is a group under multiplication.
with
E
(a,b),
as above, what sort of a curve do you get?
Let
and let
H be the s e t of ell matrices of the form
with
XI
f yK,
x E R
and
y G
R.
Prove the following:
(a)
The product of two elements of
(b)
The element XI
+yK
x
(c)
2
i s also an element o f
H.
is invertible i f and only i f
- y
The s e t of a l l elements
under multiplication.
H
2
+o.
XI
+ yK
with
x
2
- y2
= 1 is a group
94
8.
If a set
G
of
2 x 2 matrices i s a group under multiplication, show that
each of the following s e t s are groups under multiplication:
9.
where 'A
(a)
{ A ~ ;A E GI,
(b)
(8-I AB: A E G ) ,
If a a e r
G
(a)
G
(b)
G
of
=
-
2 x 2
A
(BA: A
E
= transpose of
where B
A;
is a fixed invertible element of M.
matrices is a group under multiplication, show that
E
GI,
GI,
B
where
is any fixed element o f
G.
Using the d e f i n i t i o n of an abstract group, demanstrare whether or not each
10.
of the following sets under the indicated operation is a group:
(a)
the s e t of odd integers under addition;
(b)
the s e t
(c)
the s e e of the four fourth root8 of
R+
of posririve real numbers under multiplication;
1,
1
i, 4 , under
mu1 t i p l i e a t i o n ;
(d)
the s e t of all integers of the form
3m,
where m
is an integer,
under addition.
11.
By proper application o f the four defining postulates of an abstract group,
prove that i f a, b, and c are elements in a group and a o b = a o c,
then
2-7
.
b =
c.
An Isomorphism between Complex Numbers and Efatrfces
It
i s true t h a t very many different kinds of algebraic systems can
expressed in t e r n of special collecrions of matrices.
nature have been proved in modern higher algebra.
be
Wny theorems of thir
Without attempting any such
proof, we shall aim in the preaent s e c t i o n to demonstrate how the system of
complex numbers can be expressed in term of matrices.
In the preceding section, several subaeta of the s e t of all
were d i s p l a y e d .
I n particular,
[Y"
was considered.
I:
the s e t
*
2 x 2
matrices
Z of all matrices of the form
x E R and y E R ,
We shall exhibit a one--t-ne
correspondence between the a e t
95
of
all c 0 m p l e ~numbers, which we denote by
C,
and the s e t
2.
Thf s
one-to-one
correspondence would not be particularly significant if it d i d not preserve
algebraic properties
- that
is,
if the sum of tw complex numbers b i d not
correspond to the sum of the corresponding two matrices and the product of tvo
complex numbers d i d not correspond to the product: of the corresponding two
matrices.
There are other algebraic properties that are preserved in this
sense.
Usually a complex number is expressed in the form
where
1=
6 1 ,
x
E
R,
and y
h
R.
Thus, if
c
is an element of
C,
the
s e t of a l l complex numbers, we may write
L
The numeral
is introduced in order t o make the correspondence more apparent.
In order to exhfbit: an element: of
Z
i n similar
form, we must introduce the
special matrix
Note that
thus
The matrix
J
corresponds t o the number
equation
This enables us t o v e r i f y that
i,
*
which satisfies the analogous
which indicates that any element of
Z
may be written in the form
For example, w have
and
Now we can establish a correspondence between
the s e t of complex numbers,
2 , the s e t of matrices:
and
Since each element of
of
C,
Z
C
is matched with one element of
is matched with one element o f
Z,
and each element:
C, we c a l l the correepondence o n e t r r o n e .
Several special correspondences are notable:
As s t a t e d earlier, it is interesting that there is a correspondence
between the complex numbers and
2 x 2
matrices, but the correspondence is not
particularly signiftcant unless t h e o n e t m n e matching is preserved in the
operations, e s p e c i a l l y in addition and multiplication.
We shall now f o l l o w the
correspondence in these operations and demonstrate t h a t the one-tcrane property
is preserved under the operations.
When two complex numbers are added, the real components are added, and the
imaginary components are added.
Also, remember that the multiplication of a
matrix by a number is distributive; thus, far
have
Hence we are able to show
a E il,
b
€
R,
and
A
E
M,
we
our o n p t o a n e correspondence under addition:
For example, we have
(2
- 3i)
+
(21 - 35)
( 4 f li)
= 6 - 2 1
+
(41 + 13)
=
+ (27: + OJ)
-
61-2.T.
and
(3
- 2 i ) + (2 + O i )
=5-21
(33
f-3
-
25)
51-25,
Before demonstrating that the correspondence 513 preserved under multiplication, let us review for a moment.
An example w i l l suffice:
Generally, for multiplication, we have
If
IM
represent a complex number
as a matrix,
w e do have a s i g n i f i c a n t correspondence!
Mot only is there a o n e t m n e c o r
respondence between the elements of the two s e t s , but a l s o the correspondence
is onet-ne
under the operations of addition and multiplication.
The addftive and multiplicative identity elements are, respectively,
and
and for the ~ d d tive
f
inverse of
we have
Let uil now examine how the multiplicative inverses, or reciprocals, can be
We have seen that any member of the s e t o f
matched.
2 x
2
matrices has a
multiplicative inverse i f and only i f for it the determinant function does not
equal zero.
x
2
+y
2
#
A E Z
Accordingly, if
0 , since
6(A)
= x2
+ y2
then there exlacs
for A = XI
-
+ yJ.
A
i f and only if
M u we know that any
complex number has a multiplicative inverse, or reciprocal, if and only i f the
complex nrrmber is not zero.
multiplicative inverse of
and
y
are not bath 0 .
since x
E
R
and
y E R.
That is, i f
c
c
if and only i f
x
+ yi,
then there exists a
x
+ yi #
0,
which menne that
Thls l a equivalent t o saying that x2
+y
2
# 0,
For m l t i p l i c a t i v e inverses, i f
our correspondence yields
It i a now clear that the correepoadeace between C,
nrrmbers, and
2,
a subset of all 2 x 2
matrices,
the s e t of complex
x
100
is preserved under the algebraic operations.
saying that
C
and
2
AIL o f this may be summed up by
have i d e n t i c a l algebraic structures.
Another way of
C and Z are isomorphic. This word is derived
from two Greek words and means "of the same form."
Two number systems are
expressing t h i s i s to say that
isomorphic i f , f i r s t , there is a mapping of one onto the other that i s a o n e t o -
one correspondence and, secondly, the mapping preserves sums and products.
If
t w o number systems a r e isomorphic, their structures are the same; it is only
t h e i r terminology t h a t is d i f f e r e n t .
The world is heavy with examples
morphisms, some o f them t r i v i a l and some quite the o p p o s i t e .
of
is*
One of the simplest
is the isomorphism between the counting numbers and the p o s i t i v e integers, a
subset of the integers; another fs that between the real numbers and the subset
o f a l l compLex numbers.
a 4- Oi
(We should quickly guess that there is an
isomorphism between real numbers
a1
a
and the set of all matrices of the form
+ OJ!)
An example of an isomorphism that i s more difficult to understand is that
between real numbers and residue classes of polynomials.
We won't
t r y to explain
t h i s one; but there is one more fundamental concept that can be introduced here,
as follows.
We have stated that the real numbers are isomorphic to a subset of the c
plex numbers.
m
We speak of rhe algebra of the real numbers as being embedded in
the algebra of complex numbers.
In this sense, we can say that the algebra of
2 x 2
complex numbers i s embedded i n the algebra of
A l s o , we can
matrices.
speak of the complex numbers as being "richerf1than the real numbers, or o f the
2 x 2
matrices a s being richer than the complex numbers.
The existence of
complex numbers gives us solutions to equations such as
which have no solution i n the domain of real numbers.
that
Z
i s a proper subset of
M,
that: is,
example to illustrate the statement that
M
Z
C
It is o f course clear
M and Z # M. Here i s a simple
is "richer" than
has f o r solution any matrix
X =
[l;t
:],
t t R
[see. 2-73
and
f
0,
2:
The equation
101
as may be seen quickly by easy computation; and there are still other s o l u t i o n s .
On the other hand, the equation
has exactly two s o l u t i o n s among the complex numbers, namely
c =-
c = 1
and
1.
Exercises 2-7
1,
Using the fol/loving values, show the correspondence under a d d i t i o n and
multiplication between complex numbere of the form
of the form x 1
2.
and matrices
+ yJ:
- 1, x2 = 0,
(a)
x1 = 1, y1 =
(b)
X1
(c)
x1 = 0 , y l = - 5 ,
a
x f yi
= - 2;
and
y
1,
and
y2 =
x2=3,
and
y2=4.
31 Y1 = - 4 , xZ
ti
2
1;
Carry through, in parallel coluums as in the t e x t , the necessary computations t o establish an isomorphism between
R and the s e t
by means of the correapondence
3.
In the preceding exercise, an iaomorphi~mbetween
matrices
was considered.
Define a function
f by
R
and the sets of
102
Determine which of the following statements are correct:
f l x + Y ) = f(x)
(a)
4.
Is the
with
2-8.
set
G
and
a
+ f(y1,
of matrices
b
rational and
a
2
+ b2
-
1,
a group under multiplication?
Algebras
The
concepts of group, ring, and f i e l d are of frequent occurrence i n modern
algebra.
The study o f these systems is a study o f the arructures or patterns
that are the framevork on which algebraic operations are dependent.
chapter,
r ~ ehave
In this
attempted t o demonstrate how these aame concepts describe the
structure of the
s e t of
2
X 2
matrices, which is a subset o f the s e t of all
rectangular matrices.
N o t only have we introduced these embracing concepts, but we have exhibited
the ''algebra" of the s e t s .
i n a Loose sense.
"Algebra" is a generic word t h a t i s frequently used
By technical definition, an algebra is a system that has tuo
binary operations, called "addition" and "mu1 tiplicarion," and also has
"multiplication by a number," that make it both a ring and a vector space.
Vector spaces will be discussed in Chapter 4,and we shall see then that
the s e t of
2 x 2
matrices constitutes a vector space under matrix addition
and multiplication by a number.
As you
Thus the
2
x 2 matrices form
yourself might conclude at t h i s t i m e ,
many possible algebras.
,this
an algebra.
algebra i a only one of
Some of these algebras are duplicate8 o f one another
in the aense that the b a s i c structure of one is the aame a a the basic structure
of another.
Superficially, they seem different because of the terminology.
When they have the same structure, t w o algebras are called isomorphic,
Chapter 3
MATRICES AND LINEAR SYSTEMS
3-1.
Eauivalent Systems
In this chapter, we shall demonstrate the use of matrices in the solution
of systems of linear equations.
We shall f i r s t analyze some of our present
algebraic techniques for s o l v i n g these systems, and then show how the same
techniques can be duplicated in terms of matrix operations.
Let u s begin by looking at a system of three linear equations:
In our f i r s t s t e p toward finding the s o l u t i o n s e t of t h i s system, we proceed
as follows:
M u l t i p l y Equation (1) by
Equation (1) by
-1
1
t o obtain Equation ( 1 ' ) ; multiply
and add the new equation t o Equation ( 2 ) t o obtain Equation
( 2 ' ) ; multiply Equation (1) by -2 and add the new equation to Equation (3) t o
obtain Equation ( 3 ' ) .
This gives the following system:
Before continuing, we n o t e t h a t what we have done is reversible.
can obtain System I from System I1 as follows:
to obtain Equation (1); add Equation
( 2 ) ; m u l t i p l y Equation (1')
(L' )
(2')
b h l t i p l y muation (1')
by 1
t o Equation ( 2 ' ) t o obtain Equation
by 2 and add t o Equation ( 3 ' ) t o obtain Equation ( 3 ) .
Our second etep is similar t o the first:
(1"); multiply Equation ( 2 ' )
In fact, we
Retain Equation ( 1 ' ) as Equation
by -1 to obtain Equation (2"); multiply Equation
by 3 and add the new equation to Equation (3') t o o b t a i n Equation ( 3 " ) .
This gives
Our t h i r d s t e p reverses the direction;
Mu1 t i p l y Equation (3") by - 1/8
to obtain Equation (3"' ) ; mu1 t i p l y Equation (3") by
3/8
(2") to obtain Equation (2"' ) ; multf p l y Equation (3") by
Equation ( 1 " ) to obtain Equation (1"' )
We thus get
.
x - y
and add t o Equation
1/8 and add t o
+ o =-I,
O + y f 0 = 2 ,
O + O + z = - 1 .
Now, by retaining the second and t h i r d equatione, and adding the second equation
to
the f i r s t , we obtain
or, in a more familiar
form,
In the foregoing procedure, we o b t a i n system I1 from system I, 111 from
11, fV from 111, and V from IV.
Thus
we
know that any set of values that
s a t i s f i e s system 1 must also satisfy each succeeding system; in particular,
from eystem V we f i n d that any
(K, y , z)
that s a t i s f i e s
I must be
Accordingly, there can be no other s o l u t i o n of the original system I; i f there
is a s o l u t i o n , then t h i s is it.
,
But do the values
sys terns
of linear equations
2,
-
we
have already p o i n t e d
actually s a t i s f y system
out
I?
For our
that sys tern I can be
obtained from system 11; similarly, I1 can be o b t a i n e d from 111, 111 from IV,
and I V from V.
Thus the s o l u t i o n of
V, namely
(1, 2 , - 1
Of course, you could verify by direct substitution t h a t
s a t i s f i e s I.
( 1 , 2 , -1)
s a t i s f i e s the s y s t e m I , and actually you should do this to guard against corn-
putational error.
But it is useful to note explicitly t h a t the ateps are
reversible, so t h a t the systems I, 11, 111, IV, and V are equivalent: in
accordance with the following definition:
D e f i n i t i o n 3-1.
Two systems of l i n e a r equations are said to be equivalent
if and only if each s o l u t i o n
of
either system is also a solution of the other.
We know t h a t the foregoing systems I through " are equivalc
s t e p s we have taken are reversible.
..: because the
In f a c t , the only operations we have per-
formed have been o f the f o l l o w i n g sorts:
A.
Multiply an equation by a nonzero number.
B.
Add one equation to another.
Reversing the process, we undo the addition by subtraction and the multiplication by d i v i s i o n .
Actually, there is another operation we shall sometimes perform in our
systematic solution of systems of linear equations, and i t a l s o is r e v e r s i b l e :
C.
Interchange two equations.
Thus, in s o l v i n g the system
our first step would be to interchange the f i r s t two equations in order to have
a leading
coefficient d i f f e r i n g from zero.
In the present chapter we shall investigate an orderly method of elimina-
t i o n , without regard t o the particular values o f the coefficients e x c e p t that we
shall a v o i d d i v i s i o n by
0.
Our method will be especially useful in dealing
with several s y s t e m s in which corresponding coefficients of the variables are
106
membere are d i f fereat
equal while the righ-ad
-
a situation that of ten
occurs in industrial and applied ecientific problems.
You might use the procedure, for example, i n "programming,
a method, or program, for solving a system of
'I
i .e
., deviaing
Linear equations by mane o f a
modern electronic computing machine.
Exercises S 1
1.
Solve the following ayaterrm of equation@:
(a)
(e)
+ 4y
5x + 7 y
3x
--
x+Zy+
4,
1;
z-3w-2,
w - 7 ,
y - 2 2 -
z
- 2w
u
2.
(f)
-
OX
* 0,
+ ly + oz
+OW
b,
Ox+Oy+la+Ov=c,
3;
Ox f Oy + 0 z
+ lw = d .
Solve by drawing graphs:
(b)
3x
-
-
y =
5x + 7 y
3.
-
lxfOy+Oz+Ow=a,
11,
1.
Perform the following matrix multiplications:
4. Given the following systems A and B, obtain B f r o m A by s t e p s consisting
either of w l t i p l y i n g an equation by a nonzero oonstant OF of adding an
arbitrary multiple of an equation t o another equation:
107
Are the two systems equivalent?
Why o r why not7
The solution ser of one of the following sys terns of linear equations is
j,
empty, while the other s o l u t i o n s e t contains an i n f i n i t e number o f
See i f you can determine which i s
solutions.
which, and g i v e three
particular numerical solutions for the system that does have s o l u t i o n s :
(a)
x+2y-
z=3,
x -
z = 4 ,
4x
2
.
-
y +
y f
22
=t
14;
(b)
x -I-2y
x -
4x
-
- z = 3,
y +
y
z = 4 ,
+ 22
115.
Formulation in Terms of Matrices
In applying our method t o the s o l u t i o n o f the original system of Section
3-1, namely to
we carried o u t j u s t two types of algebraic operations in o b t a i n i n g an equivalent
sys tern:
A.
Multiplication of an equation by a number o t h e r than
B.
A d d i t i o n of an equation to another equation.
0.
We noted that a third type o f operation i s sometimes required, namely:
C.
Interchange of two equations.
This third operation is needed if a c o e f f i c i e n t by which
d i v i d e is
* otherwise
would
0, and there is a subsequent equation in which the same variable
has a nonzero c o e f f i c i e n t .
The t h r e e foregoing operations can, in e f f e c t , be carried out through
matrix operations.
Before ve demonstrate this, we shall see how the matrix
notation and o p e r a t i o n s developed i n Chapter 1 can be used t o write a system
of linear equations i n matrix form and t o represent the steps in the s o l u t i o n .
Let us again consider the system w warked with in Section 3 1 :
We may d i s p l a y the detached coefficients o f
x, y,
and
as a matrix
z
A,
namely
Next, l e t
US
consider the matrices
.=[!I
the entries of
X
are the v a r i a b l e s
and
I'[
B =
x , y, z ,
members of the equations we are considering.
and of
;
B are the right-hand
By the definition o f mgtrix
multiplication, we have
which is a
3 x 1
matrix having as entries the l e f e h a n d members of the
equations of our Linear s y s rem.
NOW the equation
thar is,
is equivalent, by the definition of equality o f matrices, to the entire sys tern
o f linear equations.
It
i s an achievement not t o be taken modestly
thar re are
a b l e to consider and work with a large sys tern of equations in tern o f such a
log
s i m p l e representation as is exhibited in Equation (1).
Can you see the pattern
t h a t is emerging?
In passing, let us note that there is an i n t e r e s t i n g way of viewing the
matrix equation
where
is a given
A
3 x 3 matrix and X
Y are variable
and
3 x 1 colurrm
We recall that equations such as
matrices.
and
y
d e f i n e f u n c t i o n s , where numbers
the range.
x
=
sin x
of the domain are mapped into numbers
We may also consider Equation ( 2 ) as defining
case the domain c o n s i a t s of column matrices
matrices
Y;
X
a
and the range consists of column
For example,
the matrix equation
and a range of
3 x 1
of
function, but in this
thus we have a matrix function o f a matrix variable!
defines a function with a domain of
y
matrices
3 x 1 matrices
Thus with any column matrix of the form (3),
the equation associates a
UO
column matrix of the form ( 4 ) .
Looking again at the equation
where
is a g i v e n
A
a variable
matrices
X
3
3 x 3 matrix,
B
is a given
x 1 matrtx, we note that here we
3 x 1 matrix, and
X
have an inverse question:
( i f any) are mapped onto the particular
B?
For the case we
is
What
have
been considering,
we
found in Section 3 1 that the unique s o l u t i o n is
We
shall consider some geometric aspects
of
t h e matrix-function p o i n t of
view in Chapters 4 and 5.
We a r e
now ready to look again at
the procedure of Section 3-1, and restate
each system in terms of matrices:
[i 1 -1
The t-eaded
arrows
[I [j]"
=
indicate Lhat, as w saw in Secrion 3-1,
tho matrix
equations are equivalent,
In order t o see a little more clearly what has been done, let us look o n l y
[sec. 3-21
a t the
f ixst and last equations.
The first is
the last is
[g g g] [i] - [.i].
Our process has converted the c o e t f i c i e n r matrix
A
into the i d e n t i t y matrix
I. Recall that
Thus t o bring about this change we m u s t have completed operations equivalent
to multiplying on the l e f t by
A'.
In brief, our procedure a m u n t s
on the left by
t o multiplying each member of
A-l,
to obtain
Let ua note how far we have come.
By introducing arrays of numbers as
o b j e c t s in their own rfght, and by defining s u i t a b l e algebraic operations, we
are able to write complicated systems of equations in the simple fonn
This condensed notation, so similar to the long-familiar
which
we
Learned t o solve ''by d i v i s i o n , " indicates for us our s o l u t i o n process;
namely, multiply both sides on the left by the reciprocal or inverse o f
A,
obtaining the s o l u t i o n
This similarl ty be tween
and
should nor be pushed
too
f a r , however.
There is only one real number,
0,
that
has no reciprocal; as we already know, there are many matrices that have no
multiplicative inverses.
Nevertheless, we have succeeded in our aim, which
perhaps the general aim of mathematics :
to
i s
make the complicated simple,by
discovering i t s pattern.
Exercises 3 2
1.
Write in matrix £ o m :
(a)
4x
- 2y
f 72 = 2 ,
3x4- y + 5 z = - I ,
6y-
2.
(b) x
$
y = 2,
x - y = 2 .
z=3;
Determine the systems o f algebraic equations t o which the following matrix
equations are equivalent:
3.
Solve the following s y s t e m of equations; then restate your w r k In matrix
form:
4.
(a)
Onto what vector
map the vector
5.
Let
A =
domain
of
[;]
does the function defined by
Y
=
[i]
,
?
(b)
-
What vector
[Y~]
the function d e f i n e d by
[al a2 a) a&]
Y
,
I:[
does i t map onto the
ai, xi, y1 E R.
Discuss the
Define the inverse function, if any.
3
Inverse of a Matrix
In Section 2-2
we
wrote a system of linear equations in matrix form,
and saw that solving the system amounts t o determining t h e inverse matrix
-1
A
,
if it e x i s t s , since then ve have
whence
Our work involved a series of algebraic operations; l e t us learn how to duplicate
t h i s work by a series of matrix alterations.
TO do t h i s ,
we suppress the column
matricea in the scheme shown in Section 3 2 and look a t the coefficient matrices
u4
on the Left:
Observe what happens if ue s u b s t i t u t e "row" for "equation" in the procedure o f
Section 3-1 end repeat our steps.
In the first s t e p , we mu1 tiply row 1 by -1
and add the result t o row 2 ; multiply row 1 by -2 and add t o row 3 .
m u l t i p l y row 2 by 3 and add t o row 3; multiply row 2 by -1.
row 3 by
3/8
and add t o row 2 ; multiply row 3 by
multiply row 3 by
- L/8. b a t ;
118
we add row 2 t o row 1,
Second, we
Third, ue multiply
and add to row 1;
Through "row operations"
we have duplicated the changes i n t h e c o e f f i c i e n t matrix as w proceed through
the series of equivalent
system^.
The three algebraic operations described in Sec tlon 3-2 are peralleled by
three matrix row operations:
Definition 3-2.
The three row operations,
Interchange of any tw r o w ,
Multiplication of a l l elements of any row by a nonzero constant,
Addition o f an arbitrary multiple of any row to any othet row,
are c a l l e d elementary row operations
on
a matrix.
the exact relationship between row operations and the
In S e c t i o n 3-5,
operations developed i n Chapter 1 w i l l be demonstrated.
Earlier we d e f i n e d
equivalent systems o f linear equations; i n a corresponding way, we now define
equivalent matricee
.
Definition 3 3 .
Two matrices are said to be row equivalent i f and only if
each can be transformed into the other by means of elementary row operations.
We
now turn our attention t o the right-hand member
A t the moment, the right-hand
of
the equation
member consists solely of t h e matrix
B,
we wish temporarily t o suppress just as w temporarily suppressed the
X
in considering the left-hand member.
matrix for the right-hand member.
which
matrix
Accordingly, ue need a coefficient
To obtain this, we use the i d e n t i t y matrix
[seem
3-31
to write the equivalent equation
AX =
IB.
Now our process converts this to
which can be expressed e q u i v a l e n t l y as
When we compare EquatFon (1) and Equation ( 2 ) we notice t h a t as
I
goes to
A
A-l.
Might it be t h a t the row operation*, which convert
when applied t o the left-hand member, w i l l convert
t o the right-hand member?
Let us t r y .
goes t o
A
I,
into
I into A-Iwhen applied
For convenience, w e do t h i s in paralLel
columns, giving an indication in parentheses of how each new row is obtained:
I
4
--
L
O
O
0
0
1
left-hand Fnverse f o r
A,
8
4
4
8
8
To demonstrate that
i s a
it is necesaary to show that
are asked to verify this as an exercise, and also to check that AB = I,
-1 of A . In Section 3-5 we shall
thus demonstrating that 3 is the inverse A
You
see why i t is that
AB
-
1
follows from
BA = I
for matrices
B
that are
obtained in this way as products of elementary matrices.
We now have the f o l l o w i n g rule for computing the inverse
A,
i f there is one:
A-'
of a matrix
Find a series of elementary row operations rhat convert
I; the same series of elementary row operations
will convert the i d e n t i t y matrix I into the inverse A-'.
A i n t o the i d e n t i t y matrix
When we s t a r t the process we may n o t know if the inverse e x i s t s .
need n o t concern us at the moment.
that i s , i f
If the application of the r u l e i s successful,
then we know that
-1
s e q u e n t s e c t i o n s , w e s h a l l d i s c u s s w h a t h a p p e n s when A
A
is converted into
This
I,
A'
exists.
In sub-
doesnotexistandwe
a h a l l a l s o demonstrate that the row operations can be brought about by means of
the matrix operations rhat were developed i n Chapter 1.
Exercises 3 3
1.
Determine the i n v e r s e of each of the following matrices through row
operations.
(Check your answers.)
2.
Determine the inverse, if any, of each of the f o l l o w i n g matrices:
3.
S o l v e each of the following matrix equations:
(a'
4.
[::
5
:I [;I - [a]
1 -1
,
[: :
Solve
1 -1
x
u
m
r
3
1 - 6 -
3
5.
$1
I -1
5
Perform the multiplications
-9
-
=
[i]
,
6.
M u l t i p l y both member6 of the matrix equation
on the left by
and use the reaulr to solve the equation.
7.
Solve:
8.
Solve:
.
U9
Linear Systems of Equations
I n Sections 3 2 and 3 3 , a procedure for ~ o l v i n ga mystem of linear equations,
was presented.
exists
.
The method produces the multiplicative inverse A-I
i f it
In the p r e ~ e n tsection we shall consider systems of linear equatione in
general.
Let us begin by looking at a simple illustration.
Example 1.
Consider the system of equations,
We start in parallel eolunu~s, thus:
Proceeding, we arrive after three steps a t the following:
If we multiply these two matrices on the right by the matrices
we obtain the s y s t e m
There is no mathematical loas i n dropping the equation
0 = 0
from the system,
which then can be written equivalently as
Whatever value Is given t o
and
y
z,
t h i s value and the corresponding values of
determined by these equations s a t i s f y the original system.
x
For example,
a feu solutions are sham in the following table:
Example 2 .
Now consider the systems
If we s t a r t in p a r a l l e l columns and proceed as before, we obtain
verify)
( a s you should
Multiplying these matrices on the right by the column matrices
we o b t a i n the systems
If there were
a solution o f t h e original system, it would
this l a s t system, which is equivalent to the f i r s t .
no
tains the contradiction 0 = - 1; hence there is
Do you have
an
also be a solution of
But the l a t t e r system con-
s o l u t i o n of either system.
i n t u i t i v e geometric notion of what might be going on in
each of the above systems?
Relative to a 3dimensional rectangular coordinate
system, each of the equations represents a plane.
actually intersect in a line.
Each pair of these planes
We might hope that the three lines of intersection
(in each system) would intersect in a point.
In the f i r s t system, however, the
three lines are coincident; there ts an entire "line" of solutions.
On
the
orher hand, in the second system, the three lines are parallel; there i~
no
point that lies on all three planes.
3-2
In the example worked out in Sections
and 5 3 , the three lines intersect in a s i n g l e point.
How many possible configurations, as regarda intersections, can you l i s t
f o r 3 ~ l a n e s ,n o t n e c e s s a r i l y d i s t i n c t from one another?
They might, for example,
have exactly one point in common; o r two might be c o i n c i d e n t and the t h i r d
d i s r i n c t from but parallel t o them; and so on.
There are systems of linear
equations r b t correspond t o each o f these geometric situations.
Hers are two additional systems that even more obviously than the sys tern
in Example 2, above, have no solutions:
n u s you see
that the number o f variables as compared with the number oL equations
does n o t determine whether o x n o t there i s a solution.
[sec. 3-41
I22
The method we have developed for solving linear systems I s routine and can
be applied
ables
.
to
sys terns o f any number of linear equations i n any number of vari-
Let us examine the general case and see w h a t can happen.
have a Bystem of m
linear equations in
'If any variable occurs with coefficient
n
0
Suppoae we
variables:
in every equation, we drop it.
If
a coefficient needed as a d i v i s o r is zero, we interchange our equatlons; ve might
even interchange some of the terms in a l l the equations for the convenience of
getting a nonzero coefficient in leading position.
Otherviae we can proceed in
the customary manner un t i 1 there remain n_o equations in which any of the variable6 have nonzero c o e f f i c i e n t s , Eventually we have a syatem like this:
X1
linear
terms
in variables other t b n xl
+
... 5
5
PI,
a
B2,
tl
X2
+
and (possibly) other equations i n which no variable appears.
Either all of these other equations { i f there are any) axe of the form
in which case they can be disregarded, or there is a t least one of them of the
form
i n which caee there i r a contradiction.
If there is
a contradiction, the contradiction was present in the original
system, which means there is no solution s e t .
I23
In case there are no equations of the form
..,%
we have two po~sibilities. Either there are no variables other than xl,.
which means chat the system reducee t o
..,%
and has a unique solution, or there really are variables other than xl,,
to
which we can assign arbitrary values and obtain a family of s o l u t i o n s as i n
Example 1 , above
.
Exercises 3-4
1.
(a)
List all possible configurations, a8 regards intersections, for 3
d i s t i n c t planes
(b)
.
L i s t a l s o the addf tional possible configurations i f the plane8 are
al loved to be coincLdent
2.
.
Find the ~ o l u t i o n s ,i f any, of each of the following systeme of equations:
(a)
x
+
y
+ 22
= 1,
Z
= 3,
2x f
3x
(b)
+ 2y + 42
4
4;
x +
y +
x +
y+22=7,
y +
z-6,
2-1;
(d)
2v+
v -
4v
x +
y +
t P O ,
x + 2 y +
2-0,
- x
v-
X
+ 5y + 33 =
f
y -
1,
2 - 2 ;
l.24
y +
z + w = 2 ,
x + 2 y f
z - w = - 1 ,
2x+
(e)
4 x + 5 ~ + 3 2 - W = 0 .
3-55.
Elementary Bow Operations
three types o f algebraic operations were l i s t e d as being
In Section >I.,
involved i n the s o l u t i o n o f a linear system o f equations; in Section 3 4 , three
types of row operations were used when we d u p l i c a t e d the procedure employing
In t h i s section
matrices.
this earlier w r k ,
- in
we shall show how matrix multiplication underlies
f a c t , how t h i s basic o p e r a t i o n can duplicate the row
operations.
Let us start by looking a t whit happens if
row operations on the i d e n t i t y matrix
by a nanzero number
X
Let
represent any matrix o f this type.
J
perform any one o f the three
of order 3 ,
I
row o-f
n,
Me
we have
a
First if we mu1 t i p l y a
matrix of the form
Second, i f w e add one row of
I
another, we obtain
or one o f t h r e e other similar m a t r i c e s .
any matrix of' this t y p e .
(denoted by
L)
(What are they?)
Let
K
arand for
Third, i f we interchange tuo rows, we form matrices
o f the form
Matrices of these three types are c a l l e d elementary matrices.
D e f i n i t i o n 3-4. An elementary matrix is any square matrix obtained by
performing a single elementary raw operation on the identity matrix.
to
Each elementary matrix E ( t h a t is, each
-1
t h a t is, a matrix E
such t h a t
3, K,
or
L)
has an inverse,
For example, the inverses of the elementary matrices
are the elementary matrices
respectively, as you can v e r i f y by performing the multip~icatfonsFnvolved.
An elementary matrix is related to its i n v e r s e i n a v e r y simple w a y .
J
example,
J-1
was obtained
by multiplying a row of the i d e n t i t y matrix by
is formed by dividing the same row of the i d e n t i t y matrix by
aense
-1
J
"undoes" whatever was done to o b t a i n
conversely,
J
will undo J-l.
J
n.
For
n;
In a
from the i d e n t i t y matrix;
Hence
The product of two elementary matrices also has an inverse, as the following theorem i n d i c a res
Theorem 3-L,
and
B-',
Proof.
and
.
I f square matrices
then AB
We have
A
has an inverse
and
B
CAB)-',
of order
namely
n
have inverses
126
so that
-1 -1
B A
is the inverse o f
You will recall that, for
AB
2 x 2
by the d e f i n i t i o n of inverse.
matrices, t h i s same proof was used in
establishing Theorem 2.8.
-1
A
...,
Corollary 3-1-1.
If square matrices A , B ,
K of order
-1
-1
, B ,, ,
, then the product AB. * K has an inverse
-
.
(
n
M
a
have inverses
*K)-I,
namely
The proof by mathematical induction is left: as an exercise.
Corollary 3-1-2.
If
E1,E2,.
..,%
are elementary matrices of order
n,
then the matrix
has an inverse, namely
This follows inmediately from C o r o l l a r y 3-1-1
and the fact that each
elementary matrix has an i n v e r s e .
The primary importance of an elementary matrix r e s r s on the following
property.
If an m X n
elementary matrix
E,
matrix
A
is multiplied on the left by an
m x n
then the p r o d u c t is the matrix obtained from A
the row operation by which the elementary m a t r i x was formed from
I.
by
For
example,
You should v e r i f y these and o t h e r similar left multiplications by elementary
matrices t o familiarize yourseL f with the patterns.
Any elementary row operation can be performed on an
Theorem 3-2.
A
matrix
of order
by multiplying
A
m x n
on the left by the corresponding elementary matrix
m.
The proof is left as an exercise.
Multiplications by elementary matrices can be combined.
add
-1
times the first r o w to the second row, we would multiply
left by the product of elementary matrices of the type
Note that
in
;I'
J multiplies the first row by
-1;
A
on t h e
J-'KJ:
i t i s necessary t o multiply by
order to change the first row back agafn to its original form after
adding the f i r s t row to the second.
to
For example, to
the third, we would multiply
A
Similarly, to add
-2
times the f i r s t row
on the left by
To perform both o f the above steps at the same time, we would multiply
A
on
the left by
Since
M1
is the product of two matrices that are themselves products of
9 is the product o f elementary matrices. By Corollary
of 9 is the product, in reverse order, o f the corresponding
elementary matrices,
the inverse
3-1-1,
inverses of the elementary matrices.
Now our f i r s t s t e p in the s o l u t i o n of the system of linear equations at
the start of this chapter corresponds p r e c i s e l y t o m u l t i p l y i n g on the left by
the above matrix
left by
5'
M1.
If we multiply both sides of
System
I on page 103 on the
we obtain System 11.
For the second, t h i r d , and fourth s t e p s , the corresponding
matrix multipliers m u s t be r e s p e c t i v e l y
Thus multiplying on the left by
M2
has the effect of leaving the first row
unaltered, m u l t i p l y i n g the second row by
-1,
and adding 3 times the second
row to the t h i r d .
L e t us now take advantage o f the associative law for the nulriplication of
matrices t o form the product
We recognize the inverse of t h e orLginal c o e f f i c i e n t matrix, as determined in
S e c t i o n 3-3,
Theorem 3-3.
If
BA = I ,
where
i s a product o f elementary matrices, t h e n
B
A3 = I,
SO
that
3
-
Proof.
is the inverse of A .
By Corollary 3-1-2,
B has an inverse 3-1
BA = I
{see. 351
.
Now from
we get
whence
so that
and
Exercises 3-5
A,
1
Find matrices
2.
Express each of the following matrices as a product of elementary matrices:
(a)
1
B,
0
and
-J
[:1;
0
a
C
such t h a t
3.
Using your answers
to
Exercise 2 , form the inverse o f each of the given
matrices.
4. Find three
4 x 4
matrices, one each of the t y p e s
3,
K,
and
I,
that
w i l l accomplish elementary row transformations when a p p l i e d as a left
multiplier to a 4 x 2
matrix.
5.
Solve the following system of equations by means of elementary matrices:
6.
(a)
Find the inverse of the matrix
(b)
Express the inverse a product of elementary matrices.
(c)
Do you think the answer t o Exercise b(b) is unique?
Why or why not?
Compare, in class, your answer with the answers found by other members
of your class.
7.
Give a proof of Corollary 51-1 by mathematical induction.
8.
Perform each of the following multiplications:
m
9.
State a general conjecture you can
make on the basis of your solution of
Exercise 8.
- 5 - 3-
In this chapter, we have discussed rhe role of matrices in finding the
solutlon s e t of a system of linear equations.
We began by Looking at the
fami liar algebraic methods of obtaining such solutions and then learned to
duplica re this procedure using matrix notation and r o w operations
.
The intimate
connection between row operations and the fundamental operation o f matrix
multiplication was d e m n s t r a t e d ; any elementary operation is the reeult of l e f t
m u l t i p l i c a t i o n by an elementary matrix.
Ue can obtain the s o l u t i o n set, if it
e x i s t s , to a system of l i n e a r equations e i t h e r by algebraic methoda, o r by row
o p e r a t i o n s , or by multiplication by elementary matrices.
Each of these three
procedures involves steps t h a t are reversible, a condition t h a t assures t h e
equivalence of the sys terns.
Our work with systems of linear equationa led as to a method for producing
the inverse o f a matrix when i t exists.
that converts
a
matrix
The i d e n r i c a l sequence
o f rou operations
into the identity matrix will convert the identity
A
A, namely A-l.
The inverse is particularly h e l p f u l
when-we need to solve many systems of l i n e a r equations, each possessing the
matrix into the inverse of
same c o e f f i c i e n t matrix
but d i f f e r e n t right-hand column matrices
A
B,
Since the matrix procedure 'diagonalized' the c o e f f i c i e n t matrix, the
method is o f t e n called the "diagonalization method."
Although ve have not
previously mentioned it, there i s an a l t e r n a t i v e method for s o l v i n g a Linear
system that
is o f t e n more u s e f u l when dealing w i t h a s i n g l e system.
In this
alternative procedure, ue apply elementary matrices to reduce the system to the
form
as in Sya tern
obtained.
III in Section 3-1,
This value o f
to determine the value o f
from which the value for
z
can readily be
can then be subs ti t u t e d in t h e next to Last equation
z
y,
and so on.
An examination of the c o e f f i c i e n t
132
matrix displayed above shows clearly why t h i s procedure is called the "trtangularization method
On
."
the o t h e r hand, t h e diagonalizarion method can be speeded up to bypass
the triangularized matrix of c o e f f i c i e n t s in ( I ) , or in System 111 of S e c t i o n
3-1, altogether.
Thus, a f t e r p i v o t i n g on the c o e f f i c i e n t of
x
in the System
I,
to o b t a i n the s y s t e m
we could next p i v o t completely on
and then on the coefficient of
z
the coefficient of y
to obtain the system
to g e t the system
IV'
which you will recognize as the system V of Section 3-1,
It should now be plain that the routine diagonal i z a t i o n and triangularizat i m methods can be a p p l i e d to sya terns o f any number o f equations in any number
of
variables and that the methods can be adapted to machine computations.
Chapter 4
REPRESENTATION OF COLUMN MATRICES AS GEOMETRIC VECTORS
4-1.
The Algebra o f Vectors
In the present c h a p t e r , we shall develop a simple geometric representation
for a special class o f matrices - namely, the s e t o f column matrices
with two e n t r i e s each.
The familiar algebraic operations on this s e t of matrices
will b e reviewed and also given geometric interpretation, which w i l l lead
to
a
deeper understanding o f the m e a n i n g and implications of the algebraic concepts,
By d e f i n i t i o n , a column vector of order
2 x 1
is a
2
matrix.
Consequent-
l y , usfng the rules o f Chapter 1, we can add two such vectors or m u l t i p l y any one
of them by a number.
The s e t of column vectors o f order
2
has, i n f a c t , an
a l g e b r a i c structure with propertfes that were largely explored i n our study o f
the r u l e s o f o p e r a t i o n with matrices,
In the f o l l o w i n g pair o f theorems,
we sutmnarize what we already know con-
cerning the a l g e b r a o f these v e c t o r s , and i n the next section we shall b e g i n the
interpretation o f that algebra i n geometric terms.
Theorem 4-1.
number, and Let
Let:
A
V
and
W
be column vectors of order
be a square matrix o f order
V
are each column vectors of order
+ W,
rV,
and
2.
2,
Then
AV
2.
For e x a m p l e , i f
V =
then
and
[:I
,
W =
]:[
, r
-
4,
and
A =
[i-:]
let
r
be a
19
Ler V,
Theorem 4-2.
and
W,
A
be numbers, and let
s
U
and
B
and
be column vectors o f order
2,
let
be square matrices of order
2.
Then
r
a l l the following laws are valid.
I. Laws for the addition
11.
(a)
V + W P W + V ,
(b)
(V f
(c)
v
+ 0 = v,
(d)
V
+ ( 4 )= 2.
o f vectors:
W) + U = V
+ (W
f
U),
Laws for the numerical multiplication o f vectors:
+ W)
= rV f rW,
(a)
r(V
(b)
~(BV)= (TS)~,
(c)
(r
+ s)V
(dl
ov
= 0,
(e)
lv =
(£1 rg
a
rV
+ sV,
-
=
V,
0.
III. Laws f o r the multipLication of vectors by matrieea:
(a) A(V
+ W)
+ B)V
= AV
= AV
+ AM,
+ BV,
(b)
(A
(c)
A(BV) = (AB)V,
(dl
02V =
(e)
IV = V,
(f)
A(rV) = (KA)V = r(AV).
2,
In Theorem 4-2, 0 denotes the column vector of order
square matrix of order
2,
all of whose entries are
2,
and
O2
the
0.
Both of the preceding theorems have already been proved f o r matrices.
~'ince
column vectors are merely special types of matrices, the theorem as stared must
likewise be true.
They would a l s o be true, o f course, i f
3 or by a general
n,
of order
n
2 were replaced by
throughout, with the understanding that a column vector
is a matrix of order
n
x 1.
Exercises 4-1
1.
Let
let
r = 2
and
a=-I.;
and l e t
Verify each of the laws s t a t e d in Theorem 4-2
for this choice o f values
f o r the variables.
2.
Determine the vector
V
such that
AV
- AW = AW
+ BW,
where
3.
Determine the vector
V
such t h a t
LV
+ 2W
+ BV,
if
4.
Find
5.
Let
V,
Evaluate
i F
= AV
Using your answers to parts (a) and ( b ) , determine the e n t r i e s of
(e)
if, f o r every vector
A
and
n
theorem for
A
State
is
square matrix of
a
stands f o r any column vector o f order
V
n = 3.
Try to prove the theorem for a l l
if, for every vector
V
of order
n.
Prove rhe new
n.
2,
your result as a theorem.
Restate the theorem obtained in Exercise 7 if A
n
and
n = 3.
9.
A
Using your answers to parts (a) and (b) of Exercise 5 , determine the entries
of
8.
2,
Restate the theorem obtained in Exercise 5 if
order
7.
of order
S t a t e your r e s u l t as a theorem.
(d)
6.
V
V
stands f o r any column vector of order
T r y to prove the theorem for all
is a square matrix of order
n.
Prove t h i s theorem f o r
n.
Theorems 4-1 and 6 2 surmnarize the p r o p e r t i e s of the algebra of column
vectors with
2
entries.
S t a t e two analogous
properties of the algebra of row vectors with
theorems s u m r i z i n g the
2
entries.
'1
Show that the
t w o algebraic structures are isomorphic.
4-2.
Vectors and Their Geometric Representation
The n o t i o n of a vector occurs frequently i n the physical sciences, where a
I
v e c t o r ia o f t e n referred to as a quantity having b o t h l e n g t h and d i r e c t i o n and
accordingly i s represented by an arrow.
Thus force, v e l o c i t y , and even dis-
placement are vector q u a n t i t i e s .
Confining our attention to the coordinate plane, l e t u s invesrigate t h i s
I
i n t u i t i v e notion of vector and see how these physical or geometric vectors are
related to the algebraic column and row vectors o f S e c t i o n 4-1.
An arrow in the plane is determined when the coordinates of its &
and
i
J
the coordinates of its head are g i v e n .
is shown in Figure 4-1.
Thus the arrow
A1
from ( 1 , Z )
to
(5,4)
Such an arrow, in a given p o s i t i o n , is called a Located
I
Figure 4-1.
.
.
. .
.
. * X
Arrows in the plane.
yeetor; its r a i l is called the initial p o i n t , and its head the terminal point
(or end point), of the located vector.
h second l o c a t e d vector
and terminal p o i n t (2,5),
i n i t i a l point (-2,3)
A located vector
A
A2,
with
i s also shown in Figure 4-1.
may be described briefly by
giving f i r s t the co-
ordinates of its i n i t i a l p o i n t and then the coordinates of i t s terminal p o i n t ,
Thus the located vectors in Figure 4-1 are
A1:
(1,2)(5,4)
The h ~ r f z o n t a lrun, or
and its vertical rise, or
x
y
and
component, of
+: { - 2 , 3 ) ( 2 , 5 )
Al
is
component, is
Accordingly, by the ethagorean theorem, its length i s
a
'Its d i r e c t i o n is determined by the angle e
x
that it makes w i t h the p o s i t i v e
direction:
sin
Since
cos
sin
and
0
is the cosine of the angle that
9
makes with the
A1
are called the d i r e c t i o n cosines of A
8
y
direction,
1'
You might wnder why we d i d n o t wrlte simply
tan 8 =
instead of the equations f o r
value of
cos 8
and
2
1
T=T
s i n 8.
The reason is t h a t while the
determines slope, it does not determine the d i r e c t i o n of
tan 9
A1.
is the angle f o r t h e located vector (5,4)(1,2) opposite to A1,
Thus if
tan
fl and
but the angles
0
0
=
-2
=
12
they terminate in directions differing by
Now the
x
and
tan,;
=
from the positive
n
.
x
d i r e c t i o n cannot be equal s i n c e
components of the second located vector
y
then
A2:
(-2,3)
in Figure 4-1 are, respectively,
(2,S)
2
so that
AL
and
A2
- (-2)
have equal
= 4
x
and
5
- 3 = 2,
components and equal
y
sequently, they have t h e same length and the same direcrion.
components ; con-
They are not in
the same position, so of course they are not the same located vectors; b u t since
in dealing w i t h vectors we are especially interested in length and direction we
say that they are equivalent.
Definition 4-1.
Two located vectors are said t o be equivalent if and only
if they have the same length and the same direction.
P in the plane, there is a located vector
(and to A2) and having P as i n i t i a l p o i n t . To determine
For any prescribed p o i n t
equivalent t o
A1
u9
the coordinates o f the terminal point, you have only t o add the components o f
A1
t o the corresponding caordinates o f
P: (3,-7),
P. Thus for the i n i t i a l
point
the terminal p o i n t is
so that the located
vector i s
You might plot the initial point and the terminal p o i n t of A
3
check that it actually is equivalent t o A1
In general,
and to A 2 ,
we denote the located vector
and terminal point
in order t o
A
with f n i t i a l point
(x
l'Y1)
by
(x2,yZ)
A: (x~'Y~)(x~*Y~)
'
Its
x
components are, respectively,
and y
xZ-xL
and
Y2
- Y1-
Its length is
If r
the
0,
x
then its direceion l a determined by the angLe that i t makes w i t h
axis:
If r = 0,
we find f t convenient to say that the vector is directed In
any direction we please.
A s you w i l l s e e , this makes it possible t o s t a t e
~everaltheorems more simply, without having t o mention special cases.
example, this
null
For
vector is both parallel and perpendicular t o every direction
i n the plane!
A second located vector
is equivalent t o
as
A,
A
if and only if it has the same length and the same direction
or, what amounts to the same thing, if and o n l y if it has the same c
ponents as
m
A:
For any given p o i n t
(xO,yO),
is equivalent t o
and haa
A
the located vector
(xO,yO) as its i n i t i a l point.
It thus appears that: the located vector
is determined except for its
A
p o s i t i o n by its components
a
= x2
- X1
and
b
a=
YZ - Y l -
These can be considered as the entries of a column vector
In t h i s way, any located vector
for any given point
P,
A
determines a column vector
as the components of a located vector
locater vector
A
V
the entries of any column vector
with
A
is said to represent
A colunm vector is a "free vector''
P
Conversely,
can be considered
as i n i t i a l p o i n t .
The
V.
in the sense t h a t it determines the
components (and therefore the magnitude and direction), but:
of
V.
any Located vector that represents it.
In particular,
the column vector
a standard represen tation
-
OP : (O,O)(u,v)
[see. 4-23
we
not the p o s i t i o n ,
shall assign to
a s the located v e c t o r from the origin to the p o i n t
as illustrated in Figure 4-2;
t h i s is the representation to which we shall
ordinarily refer unless otherwise s t a t e d .
Figure 4-2.
-
Representations of the column v e c t o r
located vectors
OP
-e
and
-
Similarly, of course, the components o f the located vector
considered as the entries of a row vector.
as
QR.
A
can be
For the p r e s e n t , however, we shall
consider only column vectors and the corresponding geometric v e c t o r s ; in this
chapter, the term "vector" will ordinarily be used to mean "coLumn vector,''
not "row vector" or T'locatedv e c t o r . "
The length o f t h e located vector
is called the length or
Using the symbol
IlVlI
norm
-
OF,
to which we have previously referred,
of the column vector
to stand f o r the norm of
V,
we have
142
u
Thus, i f
and
v
are n o t both zero, t h e direction cosines of
u
and
I IVI 1
v
I IVI I
are
'
respectively; these are also called the direction cosines of the column vector
v.
The association between column or row v e c t o r s and d i r e c t e d
l i n e segments,
introduced in t h i s section, is as applicable to 34imensional space as it is to
The only difference is that: the components of a
the 2 d i m e n s i o n a l plane.
located v e c t o r in 3-dimensional space will be the entries of a column or row
v e c t o r of order 3 , not a column or row v e c t o r o f order 2 .
In the r e s t o f this chapter and i n Chapter 5 , you will see how Theorems
4-1
and 4-2
can be i n t e r p r e t e d t h r o u g h geometric operations on locared vectors
and how the algebra of matrices leads t o beautiful geometric r e s u l t s .
Exercises 4-2
1.
Of the following pairs of v e c t o r s ,
which have the same l e n g t h ?
2.
Let
V = t
t=1,
In
.
which have the same d i r e c t i o n ?
Draw arrows from the origin representing
t = 2 ,
t=3,
t = - 1,
t = -
2,
and
each case, compute t h e Length and direction cosines of
V
t = - 3 .
V.
for
143
3.
In
a rectangular c o o r d i n a t e plane, draw t h e standard representation for
Use a d i f f e r e n t c o o r d i n a t e p l a n e
each of the following s e t s of v e c t o r s .
far each s e t of v e c t o r s .
Find the length and d i r e c t i o n c o s i n e s of each
vector :
(el
,I:[ [:I,
Draw the line segments
I:[
representing
(a)
rn = 1,
b = 0;
(b)
rn = 2,
b = 1;
(c)
m =
- 1/2,
+
and
V
if
[ i] .
t = 0,
_+
1,
+ 2,
and
b = 3.
In e a c h case, v e r i f y that t h e corresponding set o f five p o i n t s
(x,y)
lies
on a line.
5.
n o
column vectors
are called p a r a l l e l provided their standard geometric
representations l i e on the same l i n e through the origin,
If
A
and
B
are nonzero parallel column vectors, de terrnine the two p o s s i b l e relations h i p s between the d i r e c t i o n c o s i n e s o f
A
and t h e direction cosines of
B.
6.
Determine a l l the vectors of the form
that are parallel to
4-3.
Geometric I n t e r p r e t a t i o n of the Multiplication of a Vector by a Number
The geometrical significance of the multiplication of a vector by a number
is readily guessed on comparing t h e geometrical representations of the vectors
V,
Zv,
and
-2V
for
By definition,
while
Thus, as you can see in Figures 4-3
V
and
2V
and 4 4 , the standard r e p r e s e n t a t i o n s of
have the same direction, while
p o i n t i n g in the opposite d i r e c t i o n .
F i g u r e 4--3.
The product o f
V
5, while those for 2V
by
2
and
is represented by an arrow
The l e n g t h o f the arrow a s s o c i a t e d w i t h
V
Figure 4 4 . The p r o d u c t o f
a v e c t o r and a negative number.
a vector and a p o s i t i v e number.
i s
-2V
-2V
each have length
10. Thus, m u l t i p l y i n g
produced a stretching o f the associated geometric vector to twice its
original Length while leaving i t s d i r e c t i o n unchanged.
Multiplication by
not only doubled the length of the arrow but a l s o reversed i t s d i r e c t i o n .
-2
These observations lead us to formulate the following theorem.
Theorem 4-3.
and Let
r
be
a
Let the d i r e c t e d line segment
number.
Then the vector
rV
the r e p r e s e n t a t i ~ nof
Proof.
-
Let
V
rV
G;
is opposite to that of
be the v e c t o r
Now,
hence,
This p r o v e s the f i r s t p a r t of the theorem.
the second part o f t h e theorem is certainly true.
the direction cosines of
-
PQ
are
[sac.
4-33
V
is represented by a directed l i n e
8. If
having length Irl times the length of
sentation o f rV has the same direction aa
represent the vector
r > 0 the repreif r < 0, the d i r e c t i o n of
G.
U
and
I IVI I
while those of
rV
ru
>
V
and
IIVII
'
are
Irl
If r
v
-
= r,
I
0, we have
and
1IVlt
rv
Irl
l lVl l
'
whence i t follows that the arrows a s s o c i a t e d w i t h
have the same d i r e c t i o n cosine^ and, therefore, the same d i r e c t i o n .
rV
If r < 0, we have
ated w i t h
Irl =
- r,
-
and the direction cosines of the arrow associ-
are the negatives of those of PQ.
rV
representation o f
rV
is o p p o s i t e t o that of
Thua, the d i r e c t i o n
G. This
o f the
completes the proof of
the theorem.
One way of s t a t i n g part of the theorem j u s t proved
a number and
V
Exercise 4 - 2 4 } ;
i s a v e c t o r , then
V
and
is t o say that if
r
is
rV are p a r a l l e l vectors (see
thus they can be represented by arrows Lying on the same line
through the origin.
On the other hand, i f the arrows representing two v e c t o r s
are parallel, i t i s easy to show that you can always express one of the vectors
as the product o f the other vector by a s u i t a b l y chosen number.
ing d i r e c t i o n cosines, i t is easy
to
Thus, by check-
v e r i f y that
are parallel vectors, and that
In the exercises that follow, you will be asked to show why the general r e s u l t
illustrated by this example holds true.
Exercises 4-3
1.
Let
L
be the s e t o f a l l vectors parallel t o the v e c t o r
rhe following blanks
so as to
[;I
.
Fill
produce in each case a true statement:
in
2.
(e)
f o r every real number
ty
(f)
f o r every real number
t,
[-"LC]
L;
Verify graphically and prove algebraically that the vectors in each of the
In each case, express the first vector as
following pairs are parallel.
the p r o d u c t o f the second vector by a number:
3.
Let
V
parallel.
4.
5.
a nonzero vector such that
W
be a vector and
Prove that there exists a real number
Prove:
(a)
If
rV=
(b)
If r v =
[:]
and
r
[i]
and
v i
V
Show that the vector
+ rV
1
4-4.
[i] ,
[:]
Y =
r
then
f rVlf
if
V
r <
= IlVll
- 1.
I1
have two vectors
V
and
W,
[sec.
V =
b3j
are
r
> - 1,
-
.
0 9
V
if
Show also that
+ rl.
Geometrical Interpretation o f the Addition of Two Vectors
If we
W
such t h a t
r
has the same d i r e c t i o n as
and the opposite direction to
IlV
then
0,
and
V
and
W =
i] ,
their sum
14a
is, by d e f i n i t i o n ,
To i n t e r p r e t the addition geometrically, l e t us return momentarily to the conc e p t of a "free" vector.
Previously we have associated a c o l u m vector
w i t h some located vector
such t h a t
c = x2
- xl and d = y 2 - y 1.
In p a r t i c u l a r , we can associate W
with the vector
A:
We use
V
+W
A
to represent
W
(a,b)(a i - c ,
b +dl.
and use t h e standard representation f o r
in Figure 4 - 5 .
Vector addition.
Figure 4-5.
A l r o , we can represent
V
as the located v e c t o r
0 : (c,d)(a
+ c,
[see.
b
4-41
+ d)
V
and
Ika
and o b t a i n an alternative representation.
If the tw p o s s i b i l i t i e s are d r a m
on one s e t of coordinate axes, we have a parallelogram in which the diagonal
represents the sum; see Figure 4-6.
Figure 4-6.
Parallelogram rule for a d d i t i o n .
The parallelogram rule is often used in physics to obtain the resultant
when
two
forces are acting from a single p o i n t .
Let us consider now the sum of three vectors
We choose the three located vectors
-
PQ : ( a , b ) ( a
and
to represent
5
V, W,
:
and
((a
+
u
C,
+ c,
b
b
+ d)(a
+ d)
-I- c
$.
e,
b f d
+-
respectively; s e e Figure 4--7.
f)
Figure 4-7.
The sum
+ W + U.
V
Order of a d d i t i o n does not a f f e c t the sum although i t does a f f e c t the
geometric r e p r e s e n t a t i o n , as i n d i c a t e d i n Figure 4-43.
Figure 4 4 .
If V
of
V
+W
and
W
The sum
V
+U
f
W.
are parallel, the construction of the proposed representative
is made i n the same manner.
Theorem 4-4. If the vectors
V
The d e t a i l s w i l l be l e f t to the s t u d e n t .
and
W
[sec. 4-41
are represented by the directed
-
Ija
-L
rr,
line segments
Since
OP
and
V -W = V
V .t W
respectively, then
PQ,
+ (-W),
is represented by
OQ.
the operation of subtracting one vector from
another o f f e r s no essentially new geometric idea, once the construcrion o f
is undets t o o d
representtng
.
It is useful to note, however, that since
the Length of t h e v e c t o r
P:
(u, v ) and
lllus trates the construction of the geome e r i c vector
Figure 4-9
V - W.
T: (r,
V
-W
equals the d i s t a n c e between the points
s).
T*
Figure 4-9.
The subtraction
/
of
T: (r, a )
vectors,
V
- W.
Exercises 4 4
1.
Determine graphically the sum and difference
vectors.
+
of the following pairs of
Does order matter in cons tructfng the sum?
the difference?
152
2.
Illustrate graphically the associative law:
3.
Compute each of the following graphically:
4.
State
5.
the geometric significance of the following equations:
I:[
(b)
V + W + U P
(c)
V + W f U + T =
*
I:[
Complete the proof of b o t h parts a f Theorem 4-4.
4-55, The Inner Product of Two Vectors
Thus f a r in our development, we have i n v e s t i g a t e d a geometrical interpret a t i o n for the algebra of vectore.
We have represented column vectors of order
2 as arrows in the plane, and have established a one--twne correspondence b e
tween t h i s s e t of column vectors and the set: of d i r e c t e d l i n e segments from the
origin a f a coordinate plane.
The algebraic operationa o f a d d i t i o n of two
vectors and of multiplication of a vector by a number have acquired geometrical
s Fgni ficance
.
But we can a l e o reverse our point of view and see that the geometry of
vectors can lead us to the consideration o f a d d i t i o n a l algebraic structure.
For instance, if you look at the p a i r of arrows drawn in Figure 4--10, you
may
c m e n t that they appear t o be mutually perpendicular.
You have begun t o
Figure 4-10.
Perpendicular vectors.
talk about the angle between the pair of arrows.
Since our vectors are located
vectors, the following d e f i n i t i o n is needed.
Definition 4-2.
The angle between tm, vectors i s the angle between the
standard geometric representations of the vectore
Let us suppose, in general, t h a t the points
and
R,
w i t h coordinates
8,
POR.
If
Consider the angle
in the triangle
We can compute the cosine of
triangle
P, w i t h coordinates (a,b) ,
are the terminal points of two geometric
(c,d),
v e c t o r s with initial points a t the o r i g i n .
denote by the Greek letter
.
0
IOPI , IORI , and
WR
by applying the Law of cosines to the
lPRl
are the lengths o f the sides of
The angle between two vectors.
I-.
which we
of Figure 4-11.
the triangle, then by the law of cosines we have
Figure 4-11.
WR,
4-51
Thus,
= 2(ae f. bd)
.
Hence,
IOPI IORI cos 61 = ac
+ bd.
(1l
The number on the right--hand side of this equation, although clearly a function
of the two vectors, has n o t heretofore appeared explicitly.
L e t us give it a
name and introduce, thereby, a new binary operation f o r vectors.
Definition 4-3.
The inner product of the vectors
is the algebraic sum of the products of corresponding entries.
]
[;]
-
sc
Symbolically,
+ bd.
We can similarly define the inner product of cur row vectors:
ac
+ bd.
[ a b] rn
[= d]
Another name for the fnner product of two vectors is the "dot product" of
the vectors.
number.
say
You n o t i c e
that the inner product of a pair of vectors is simply a
In Chapter 1, you met the product of
[ab]
times
[i] ,
and found that
a row vector by a column vector,
-
w
the product being a
1 x 1 matrix.
As you can observe, these two k i n d s of
products are closely related; for, i f
[:]
and
[
,
vt = [a
w have
vt"
=
V
and
are the respective vector8
and
b]
[ae
W
-
+ bd]
[V
w] .
Later we shall exploit this close connection between the ttro product& i n order
ro deduce the algebraic properties of the inner product from the known properties
of the matrix product.
Using the notion of the inner product and the formula (1) obtained above,
we can s t a t e another theorem.
We shall epeak
o f the
cosine of the angle included
between two column (or row) v e c t o r s , although we realize that we are actually
referring to an angle between aseociated directed l i n e segments.
Theorem 4-5.
The inner product of two vectors equals the product o f the
lengths of the vectors by the c o s i n e o f t h e i r included angle.
V
where
0
.
w
=
l lVl l
i s the angle between the veetore
Theorem 4--5 has been proved in the case
l l W l l cos 8 ,
V
W.
in which V
and
and
W
a r e not
If we agree t o take the measure of the angle between tul
parallel vectors.
parallel vectors
Symbolically,
to
be '0
or
180'
according as the vectors have the same or
opposite directions, the result s t i l l holds.
Indeed, as you may recall, the law
of cosines on which the burden of the proof rests remains valid even when the
three vertice~of the "triangle"
Collinear vectors
in the same direction.
Figure 4-12.
POR are collinear ( ~ i g u r e s4-12 and 4-13).
Figure 4-13. Collinear vectors
in o p p o s i t e direc t i o n e .
Iss
The relationship
CoroLLary 4-3-1.
holds for every vector
V.
The corollary follows a t once from Theorem 4-5 by t a k i n g
case
8 =
V =
]
,
we have
2
while
v m V = a 2 + b ,
Two column vectors
and
in which
To be sure, the result a l s o follows iwnediately from the facts
.
'
0
t h a t , f o r any vector
OP
V = W,
V
and
W
IIVII =
d m .
are said to be orthogonal i f the arroua
them are perpendicular to each o t h e r .
OR
is orthogonal to every vector.
the null vector
cos
90'
In particular,
Since
= cos 270'
= 0,
V
are orthogonal i f and only i f
we have the following result:
Corollary 4-+2.
The vectors
You will note that the condition
i f either
We
V
or W
and
W
V a W =0
i s automatically s a t i s f i e d
i s the null vector.
have examined 8ome of the geometrical facets o f the inner product of two
vectore, b u t Let us now look a t same o f i t s algebraic properties.
Does i t sat-
i ~ f yconarmtative, associative, or other algebraic laws we have met i n studying
number aystems?
We can show that the ctmrmurative Law h o l d s , that is,
For if V
and W
are any pairs of 2 x 1 matrices, a computation shows that
But
V% =
W] , while
[V
W'V
-
[W
.
.
V]
Hence
It is equally poseible
products.
Indeed, the products
To evaluate V
a (W
o f the vector
V
(W
U)
and
(V
W)
U
are meaningless.
W
U.
But the inner product is defined for
column vectors and not for a vector and a number.
two
V(W
a
U)
Inci-
should n o t be confused with the meaningless
The former product has meaning, for it is the product of the vector
by the number W
V
4
U) , for example, you are aaked t o find the inner product
dentally, the product
(W a U).
V
wlth the number
two row vectors or
V
t o show that the aaaociative Law cannot hold for inner
U.
In the exercises that follow, you will be asked t o coneider gome o f the
In particular, you will be asked
other p o s s i b l e properties of the inner product.
ro establish the following theorem, the f i r s t part of which was proved above.
If V,
Theorem4-6.
W,
D are column vectors of order 2,
and
and
is a r e a l number, then
(a)
V m W n W o V ,
(b)
(rV)
(d)
V a
V
(e)
if
VeV-0,
W
a
2 0;
r(V a W),
and
then V -
Exercises 4-5
1.
Compute t h e cosine
following pairs:
o f the angle between the two vector6 i n each of the
r
In which cases, if any, are the vectors orthogonal ?
In which
cases,
if any,
are the vectors parallel?
2.
Let
ELz[:]
and
Show that, for every nonzero v e c t o r
V
.
3.
(a)
Prove that
two vectors
[:I.
V,
V • EZ
El
and
1 IVI 1
are the direction cosines of
E2=
I IV1 I
V.
V
and
W
are parallel if and o n l y i f
Explain the significance of the s i g n of the right-hand side o f t h i s equation.
(b)
Prove that
and write this inequality in terms of the entries of
(c)
Show also t h a t
V
W
<
-
V
and
W.
llVll IlWll.
4.
Show t h a t i f
5.
Pill in the bLanks i n the following statements s o a s to make the resulting
V
is the null v e c t o r then
sentences true:
(a)
The vectors
[:] I'!-[
and
are parallel.
(b)
The v e c t ~ r s
[:I I : [
and
are orthogonal.
The vectors
and
are
The vectors
and
are para1 le 1.
For every p o s i t i v e real number
(e)
[
(f)
and
[]
end
the vectors
are orthogonal.
For e v e r y negative real number
[:I
t,
[:]
t,
the vectors
.
are orthogonal
6. Verify that parts (a) - (d) of Theorem 4-6 are true if
7.
Prove Theorem 4 - 4
(a)
by using the definition of the inner product of two vectors;
(b)
hy using the fact t h a t the matrix product
V%
s a t i s f i e s the
equation
8.
+2
Show t h a t , in each of the following s e t s of vectors,
V
I IV
+ Wl l 2
every pair of vectors
9.
-
= l lVl iZ
Pmve t h a t
orthogonal,
V
and
V
(V
+ W)
(V
+ W)
and W.
T are p a r a l l e l , and
Do t h e same relationships hold for the s e t
T
and
W
V
.
W
and
+
W
l lUl l Z for
are
are orthogonal:
160
10. Let
V
T and
11.
b e a nonzero v e c t o r .
V
are parallel.
Suppose that
Show that
W
12. Let
V
=
[z] ,
[-:I
and
r,
Show that the v e c t o r s
14.
Show that i f
A =
V
[:]
and
I IAI i Z
15.
Show that the vectors
16.
Show that the equation
holds f o r a l l vectors
17.
I
[i] ,
B =
I B Il 2
- (A
V
and
W
V
and
W.
t
holds for all v e c t o r s
V
+ WII
and
W.
the v e c t o r s
Show t h a t if
such that
8)
2
then
= (ad
- bc) 2 .
are orthogonal if and o n l y i f
Show t h a t the inequality
IIV
t,
are orthogonal if and only if
W
and
are orthogonal, while
are orthogonal.
are orthogonal, there exfs ts a real number
13.
V
and
a,
is n o t the zero v e c t o r .
V
where
and
T are then orthogonal,
and
Show t h a t , f o r every s e t of real numbers
[i]
W
<
-
IIVll
+
IlWll
W and
V
4-6.
Geometric Considerations
In S e c t i o n
That is, if
4 4 , we saw that two parallel v e c t o r s determine a parallelogram.
.A
u
I:[
and
8 =
[i]
are t w o nonparallel vectors w i t h i n i t i a l p o i n t s a t the origin, then the p o i n t s
Pt(a,b),
0:(0,0),
R;(c,d)
and
S:(a + c ,
parallelogram ( F i g u r e 4-14.)
determine the area of the parallelogram
R:
b +d)
are the vertices of a
A reasonable question to ask is: '!How
(c,
ME?"
d)
A parallelogram determined by
Figure 4-14,
can we
vectors.
As you recall, the area of a parallelogram equals the product of the lengths
Thus, in Figure 4--15, the area of the parallel*
of its base and its a l t i t u d e .
gram
is
of the altitude
blh,
where
bl
is the length of s i d e
NM and h
i s the length
KD.
Figure 4 1 5 .
Determination of the area of a parallelogram.
[see, 4-61
162
~ u ti f
NKL
angle
b2
is the length of side
or a n g l e
NK,
we have
KNM,
blb2 l s i n 8 1
Hence, the area o f the parallelogram equals
Returning to Figure 4-14 and letting
A
and
B,
is the measure of e i t h e r
8
and
we can now say t h a t if
G
2
.
be the angle between t h e vectors
0
1s the area of parallelogram
G
= I t*,i2
I IBI I
2
PORS,
2
sin 8 .
Now
sin
2
= 1
- cos2
e.
It follows from Theorem 4-5 that
COS
2
e
(A
=
.
B)
r:
I IA1 l 2 I I B I I
2
'
therefore,
Thus, we have
G~
= 1 1 ~ 1 1 1~ ~ 1 1 ~(A.
B)
2
.
It follows from the result of Exercise 14 o f the preceding section that
'G
= (ad
- bc) 2 .
Therefore,
G = lad
- bcl.
then
E
But
~trix
f
theorem,
ad
- bc
[ E] .
L"
is the value of the determinant 6(D),
where D I s the
For easy reference, l e t ue vrtte our r e s u l t in the form of a
"1
Theareri 4-7.
The area o f the parallelogram determined by the standard
;I.
representation of the vectors
D =
[;
Corollary 4-7-1.
if S(D) =
equals
The vectors
16(D)I,
where
are parallel if an? only
0.
The argument proving the corollary is l e f t
at3
an exercise f o r the s t u d e n t .
You n o t i c e that we have been l e d t o the determinant of a
examining a geometrical interpretation o f vectors.
2 x 2
matrix in
The role of matrices in t h i s
interpretation will be further investigated in Chapter 5.
From geometric considerations, you know that the cosine of an angle cannot
exceed 1 in absolute value,
and that the length of any side o f a triangle cannot exceed the sum of the
lengths of the other two sides,
+ PQ.
OQ 5 OF
Accordingly, by the geometric interprets t i o n of column v e c t o r s , for any
V
- [;]
and
Y
- [;]
we must have
i
and
IlV
+ Wll
5 lIV11 +
[see.
4-61
IlWlI.
164
But can these i n e q u a l i t i e s be established algebraically?
L e t us s e e .
The i n e q u a l i t y (1) is equivalent to
(V
. . .
w)~<
(V
V)(W
W),
t h a t is,
or
- as
we see when we m u l t i p l y our and simplify
- to
But t h i s can be written as
0
5 (ad
- bc) 2 ,
which certainly is v a l i d s i n c e the square of any real number is nonnegative.
Since
ad
- bc
= 6(D),
where
you can see that the foregoing result is consistent w i t h C o r o l l a r y 4-7-1,
that i a , the sign of equality holds in (I) if and only i f the vectors
W
are p a r a l l e l .
As for the inequality ( 2 ) , i t can be written as
which simplifies t o
which a g a i n is valid since
0
<
- (ad
- bc)".
Is-.
b63
above;
V and
165
This rime, the s i g n of equality holds if and only if the vectors
V
and
w
are parallel and
that is, i f and only i f the vectors
V
and
W
are parallel and in the same
direction.
If you would like t o look further i n t o the study of inequalitiee and their
applications, you might consult the SMSG Monograph
"An Introduction t o
Inequalities," by E. F. Beckenbaeh and R. hllman.
1.
Let
-
OP
Exercises 4 4
repreeent the vector
A,
end
6?
the vector
if
area of triangle TOP
2.
Compute the area o f the triangle w i t h vertices:
3
Verify the inequalities
(V.
and
for the vectors
wl2
< (V.
V)(Y.")
B.
Determine the
4-7.
(a)
V = (3,4)
and
W = (5,121,
(b)
V = (2.1)
and
W
=
(c)
V = (-2,-1)
and
W
= (4,Z).
(4,2),
Vector Spaces and Subapacea
Thus f a r our discussion of vectors has been concerned essentially u i t h
individual vectors and o p e r a t i o n s on them.
In t h i s section we shall take a
broader point of view.
It will be convenient
t o have a symbol for the s e t of
2 x 1
matrices.
Thus we let
where
R
L s the s e t o f real numbers.
The s e t
H
together u i t h the operations
of addition of vectors and of multiplication of a vector by a real number is an
example of an algebraic system called a vector space.
D e f i n i t i o n 4-4.
A s e t of elements is a vector space over the s e t
R
of
real numbers provided the following conditions are a a t i s f i e d :
( a ) The sum of any tw elements of the set is aLao an element of the
set.
(b)
The product of any element of the s e t by a real number is also an
element of the set.
(c)
The Laws I and If o f Theorem 4-2 h o l d .
In applying laws I and I f ,
space,
0 will denote the zero element of the
vector
Let us emphasize, houever,that: the elements of a vector space are n o t
necessarily vectors i n the sense thus far discussed in this chapter; f o r example, the s e t of
2 x 2
matrices, together w i t h ordinary m a t r i x addition and
multiplication by a number, forms e vector space.
Since a vector space consists of a s e t together with the operations of
addition of elements and of multfplication o f elements by real numbers, strictly
speaking ue should n o t use the same symbol for the set o f elements and for the
vector space.
But the practice i s not likely t o cauae confusion and will be
[sea.
4-63
167
£0 1 lowed.
A completely trivial example of a vector space over
sisting of the zero vector
set
It
of vectors parallel
is
alone.
Another v e c t o r space over
that is, t h e
to
is the set con-
R
is the
R
set
evident t h a t we are concerned uith subsets o f
H
in these two examples.
Actually, these subsets are subspaces, in accordance with the following de finition.
D e f i n i t i o n 4--5, Any nonempty subset
F
of
H
H
is a subspace of
provided the following conditions are s a t i s f i e d :
The sum o f any two elements o f
is a l s o an element of
F
by a real number i s an element o f
F
The product o f any element o f
F.
By definition, a subspace must contain a t least one element
rV f o r real numbers r .
therefore has the zero vector as an element,since f o r r
m u s t contain each
H
V
F,
and also
Every subspaee a f
of the produces
=
0 we have
It is eeay to see that the set consisting of the zero vector alone is a subspace.
We can a l s o verify that the s e t of a l l vectors p a r a l l e l t o any given nonzero
vector is a subspace.
Other than
H
i t s e l f , subsets o f these two types are
the only subspaces of H.
Theorem 4 - 8 .
Every subspace of
H consists of e x a c t l y one o f the folio*
ing: the zero vector; the set of vector^ p a r a l l e l to a nonzero vector;
space
H
itself,
Proof.
If
P
is a subspace containing only one vector, then
since the zero v e c t o r belongs to every subs pace.
[see,
b73
the
168
contains a nonzero vector
F
ff
f o r real
V,
Accordingly, if a l l vectors of
r.
F
then
F
contains all vectors
rV
are parallel t o
V,
i t
V,
F
is actually
follows
that
If F also contains
equal to
a vector
W n o t parallel t o
then
H, as we shall now prove.
Let
be nonparallel vectors in the subspace
be any other v e c t o r of
H.
F,
and l e t
We shall show that:
Z
is a member o f
F.
By the d e f i n i t i o n of subspace, t h i n will be the case if there are numbers
x
and
such that
y
that is,
These can be found i f we can solve the system
for
x
since
ad - bc
and
V
and
+O
in terms of the known numbera,
y
W
a , b, e , d ,
r,
and
s.
are n o t parallel, i t follows (see Corollary q - 1 ) that
and therefore the equations have the solution
But
169
Since
is a subspace that contains
F
their sun
Thus every vector
2.
a subset of
F.
F
But
2
and
V
of
H
it contains
W,
must belong to
xV,
yW,
F ; that is,
H. Accordingly,
is given to be a subset of
and
H
F
=
is
H.
Using the ideas o f Section 4 4 , we can g i v e a geometric i n t e r p r e t a t i o n t o
Equation (1) .
L e t the nonparallel v e c t o r s
standard representations
i%
OT.
be represented by
t o one of them must
through
T
-OP
and
4
Since
V
and
W
in the subspace
O?i,
respectively; s e e Figure 4-16,
and
O?R
5
and
d,
have
Let
2
are n o t parallel, any line parallel
intersect the line containing the other.
p a r a l l e l to
F
and let
S
which these l i n e s i n t e r s e c t rhe lines containing
-
and
OR
(1
and
Draw the l i n e s
-
be t h e p o i n t s in
OP
respectively.
Then
Figure 4-16. Representation o f an arbitrary
vector Z a s a linear combination of a given
pair of nonparallel v e c t o r s V and W.
But
x
is parallel to
and
y
6$
to
OR.
Therefore, there are real numbers
such that
6?j
Hence,
4
and
=
ZP and
d
4
OS = yOR.
This ends our discussion
of
Theorem 4-7 and introduces the important concept
o f a linear combination:
If
Definitfon 4--6.
where
x
and
y
a vector
can be expressed in the form xV
Z
are real numbers and
caLled a linear combination
V
of
and
V
and
W
are vectors, then
+ yW,
2
is
W.
Further, we have incidentally established the useful f a c t s s t a t e d in the
following theorems:
Theorem 4-9.
pair of vectors in
Theorem 4-10.
A subspace
F
contains every linear combination of each
F.
Each vector of
H can be expressed as a linear combination
of any given pair of nonparallel vectors in
H.
For example, t o express
as a linear combination of
we must determine
real numbers
x
and
y
such that
Thus, we must solve the s e t of equations
We f i n d the unique s o l u t i o n
x = 2
and
y = 1;
that is, we have
X f you observe rher the given vectors
W = 0
V
are orthogonal, that is,
V
(see Corollary 4-5-2
second method of s o l u t i o n may occur ro you.
then for the products
V
2
V =
Z
and
allVll
in the foregoing example
W
and
Z
For if
you have
W
2
on page 156) then a
and
W = bltWll
Z
2
.
But
V a 50, Z
Z
W = 25, 1 1 ~ 1 =1 ~25, and
-
IIW112
25.
Hence,
50 = 25a and
,
25 = 2%
and accordingly
a = 2
It
bSl.
and
i s worth noting t h a t the representation of a v e c t o r
combination of two given nonparallel vectors
Z
as a Linear
unique; that i s , if the vectors
is
V
and W
y
can be chosen in exactly one way (Exercise 4--7-11, below) so that
are not p a r a l l e l , then for each vector
The pair o f nonparallel vectors
real numbers
basis
.
x and y
V
and
W
the c o e f f i c i e n t a
Z
is called a basis for
are called the coordinates of
\ li]
r
In the example above, the vector
Z
x
and
H, while the
r e l a t i v e to that
C l
has coordinates
2
and
1
reLative to the basis
In particular, the pair of vectors
natural basis for
point
(u,v)
H.
and
[y ]
1s called the
This basis allows us to employ the coordinates of the
associated with the vector
t o the basis; thus,
[i]
V =
[:I
as the coordinates relative
Since every vector of
H
can
be expressed as a linear combination of any
pair of basic vectors, t h e basis vectors are said to span the vector space.
The minimal number of v e c t o r s t h a t span s vector space is c a l l e d the dimension
of the particular space.
For example, the dimension of t h e vector space H
sense, the s e t
is
2.
In the same
F
is a subspace of dimension
f o r this subspace.
Note that neither
1.
I:[
nor
[;I
is a baais
(What is?)
In a 2-space, t h a t is, a vector space o f dimension 2 , it i s necessary that
any s e t of b a s i s vectors be linearly independent,
Two vectors
D e f i n i t i o n 4-7.
only i f , f o r a l l real numbers
x = y = 0.
implies
Otherwise,
and
W
V
are linearly dependent.
w h i c h indicates that
and
and
y,
and
are linearly independent f f and
W
the equation
W
are said to be l i n e a r l y dependent.
V =
For example, let
V
x
V
V
and
W
Note that
are parallel,
Exercise 4-7
1
i
1.
Express each o f the following v e c t o r s as linear combinations of
,
i
1
2.
and i l l u s t r a t e your answer g r a p h i c a l l y :
In parts
[-:I
[:I
and
through ( i ) of Exercise 1 , determine the coordinates o f each
(a)
of the vectors relative to the basis
and
[:].
3.
Prove t h a t the following s e t i s a subspace of
H:
.
Prove that, for any given vector
[rW : r E R]
W,
the s e t
is a subspace
of H.
5.
I
(a)
I-t
I
- I:[
,
determine which o f the following subsets of
I
I6
For
all V
with
u = 0,
(b) a l l
(c)
6.
Prove that
Show that
(d)
all
V
with
V with v
to an integer,
equal
(e)
all
V
with
all
rational,
(f)
all
V
with
F
V
with
u
is a subspace o f
i f and only if
H
linear cmnbination of two vectors in
7.
H are subspacee:
F
2u
- v = 0,
u + v = 2,
uv = 0.
contains every
F.
cannot be expressed as a l i n e a r combination of the
vectors
!
8.
t
Describe the s e t of a l l linear combinations o f two given parallel vectors,
[aec.
4-73
9,
F1
Let
F2 be subspaces of
and
In proving Theorem 4-10,
Prove t h a t the s e t
F of all
F 1 and F2 is a l s o a subspace.
v e c t o r s belonging to both
10.
H.
we showed that:
vectors, then each vector of
if
V
and
W
are n o t parallel
H can be expressed as a linear combination
If each vector of H has a representation as a linear combination of V and W, then V and W are not
of
V
and
W.
Prove the converse:
parallel.
11.
Prove that i f
vector
a
12.
and
V
and
in the form
b
W
are not parallel, then the representation o f any
aV
+ bW
i s
unique;
that i s , the coefficients
can b e chosen in exactly one way.
Show that any vector
can be expressed uniquely as a linear combination of the basis vectors
4-8.
Summary
In this chapter, we have developed a geometrical representation - namely,
d i r e c t e d line segments
- for
2 x 1 matrices, or coluum vectors.
Guided by
the definition o f the algebraic operation o f addition o f v e c t o r s , we have found
the "parallelogram law of additiont' of directed Line segments.
The multiplica-
t i o n of a vector by a number has been represented by the expansion or contraction
o f the corresponding directed line segment by a factor equal t o the number, w i t h
the sign of the factor determining whether or not the direction of the Line
segment is reversed.
Thus, from a set of algebraic elements we have produced a
s e t of geometric elements.
Geometrical observations in turn led us back to
additional algebraic concepts.
Also in this chapter, we have introduced the important concepts of a vector
space and l i n e a r independence.
Since the nature of the elements of a vector space i s not limited except
by the postulates, the vector space does not necessarily c o n s i s t of a set whose
175
elements are
2 x 1
column v e c t o r s ; thus i t s elements might be
n x
matrices,
n
real numbers, and so on.
P of linear and constant polynomials
For example l e t us look at the s e t
with real coefficients, that is, the s e t
under ordinary a d d i t i o n and multiplication by a number.
The sum of two such
polynomials,
is an element oi t h e s e t s i n c e the sums
The product of any element
of
P
+ a2
and
bl
+ b2
are real numbers.
P by a real number,
c(ax
is also a member of
a1
+ b)
= acx
since the products
+ bc,
ac
and
bc
are real numbers.
We
can similarly show that addition is c o m u t a t i v e and as~ociative,t h a t there is
an i d e n t i t y for addition, and that each element has an additive inverse; thus,
the laws 5: o f Theorem 4 . 2 are v a l i d .
In l i k e f a s h i o n , we can demonstrate that
laws I1 are s a t i s f i e d : both d i s t r i b u t i v e l a w s hold; the mu1 tiplication of an
element by two real numbers is associative; the product of any element
the real number
1
is
p
itself; the product of
0 and any element:
by
p
p
is the
zero element; and the product of any real number and the zero element is the
zero element.
We have outlined the p r o o f t h a t the s e t
nomials is a vector space.
P
of linear and constant poly-
Thus the expression, "the vector,
meaningful when w e are speaking of the vector space
ax
+ by"
is
P.
The mathematics t o which our algebra has led us forms the beginnings of a
discipline called "vector analysis," which i s an important t o o l in classical
and modern p h y s i c s , as we11 as in geometry.
The "free" vectors t h a t you meet
in physics, namely, f n r c e s , velocities, etc., can be represented by our geometric
vectors.
The study i n which we are engaged i s consequently of v i t a l importance
f o r physicists, engineers, and other a p p l i e d scientists, as well as for math-
maticians
.
Chapter 5
TRANSFORMATIONS OF THE PLANE
Functions and Geometric Transformations
5-1.
You have discovered that one o f the most fundamental concepts i n your study
of mathematics is the n o t i o n of
a function.
appears i n the idea of a transformation.
In geometry the function concept
11: is the a i m o f this chapter t o recall
what we mean by a function, to define geometric transformation, and to explore
t h e role of matrices in the study of a significant c l a s s of these transformations.
You r e c a l l that a function from s e t
mapping from the elements of t h e s e t
element of A
A
B
to set
is
a correspondence or
there is associated exactly one element of
the domain of the function and the subset of
-
B
t o those ofathe set
A
B
The s e t
0.
onto which
such that with each
A
A
is
is mapped is the
In your previous work, the functions you net generally
range of the function.
had s e t s o f real numbers both for domain and f o r range.
Thus the function
s p b o l i z e d i n the form
is likely to be interpreted as a s s o c i a t i n g the nonnegative real number
the r e a l number
x.
xZ
v i th
Here you have a simple example o f a "real function'' of a
"real v a r i a b l e .I1
In Chapter 4 , however, you m e t a function
domain the vector space
t lVl
V
H, and f o r its range the
having f o r its
s e t of nonnegative
real
numbers.
In the present chapter, we shall c o n s i d e r functions rhar have their range
as well as their domain in
H.
SpecificeLly,
we want to
find a geometric in-
terpretation for these "vector functions" of a "vector variable"; this is a
-
continuation of the discussion started in Chapter 3.
considered i n their standard representations
OP,
A l l v e c t o r s will now be
so that they w i l l be p a r a l l e l
if and only if represented by c o l l i n e a r geometric v e c t o r s .
Such a vector function will associate, with the point
(x,y),
a point
geometric vector
-
P'
OP
with coordinate&
x
y
d
onto the geometric vector
OP'
.
Pf
of t h i s plane.
having coordinates
The function can, t h e r e
fore, be viewed as a process that associates with each point
same point
P
Or we may say that it maps the
P of the plane
W
e shall c a l l this process a transformation of
the plane into i t s e l f or a geometric transformation.
As a matter of f a c t ,
these transformations are of ten called "point transformations" i n conrras t t o
more general mappings i n which a p o i n t may be carried into a l i n e , a c i r c l e , or
For us, a geometric transformation i s a
some other geometric c o n f i g u r a t i o n .
helpful means o f visualizing a vector function of a vector variable.
As a
matter of c o n v e n i e n t terminology, we s h a l l call the vector that such a f u n c t i o n
a s s o c i a t e s with
a given
vector
the image of
V
V;
furthermore, we s h a l l say
9 V ontD its image.
that the function
Let us look a t t h e simple function
This function maps each vector
as
V,
V
onto the vector that has the same d i r e c t i o n
V.
but: that is twice as long as
Another way of asserting t h i s i s to
say that the function associates with each p o i n t
such that
P and
see Figure 5-1.
P'
P
of the plane a point
Pt
l i e on the same ray from the origin, but
You may therefore think
o f the function
uniformly stretching the plane by a factor
i n this example as
in a l l directions from t h e origin.
2
(Under this mapping, what is the point onto which t h e origin i s mapped?)
As a second example, consider the function
This time, each
vector i s mapped onto
the vector having length equal and direction
opposite to that o f the given vector.
function associates with any point:
P
Viewed as a point transformation, the
its "reflection" i n the o r i g i n ; see
Figure 5-2.
The function
combines both of the effects
associated with
that of
V.
V
of
the preceding functions, so that the v e c t o r
is twice as Long as
V,
b u t has the o p p o s i t e direction to
Figure 5-1.
Figure 5 2 .
The rransformation V -9-V.
The trans-
V
formation
-+ 2V.
Now, let us look a t the function
As in our first example, each vector is mapped by the function onto a vector
having the same d i r e c t i o n as the given v e c t o r .
1
is its own image.
greater than that of
of
V
itself.
having length
But if
V,
> 1,
V
then the image of
is mapped onto
which is twice as long.
whose length i s 13,
V
has a length
with the expansion factor increasing with the length
Thus, the vector
2,
Indeed, every vector of length
The vector
has the image
[sec. 51)
w i t h length 1 6 9 .
On t h e other hand, for nonzero vectors of length less than
1, we obtain image vectors of shorter length, t h e contraction factor decreasing
w i t h decreacing length of the original v e c t o r .
Thus,
is mapped onto
the Lmaze being half as long as the given v e c t o r .
Again, the vector
t h e length of the first vector being 5/7, while the length of i t s image i s
2
only ( 5 / 7 ) , or 2 5 / 4 9 . Although we may try to think o f t h i s mapping as a
kind of stretching o f the p l a n e i n a l l d i r e c t i o n s from the o r i g i n , so that any
point and its image are collinear with the o r i g i n , t h i s mental picture has also
to take into account the fact that the amount of expansion varies w i t h the
distance o f a given point from the origin, and that f o r p o i n t s within the circle
of radius
about the origin the s e c a l l e d stretching is actually a compression.
1
We have been considering transformations of individual vectors; l e t us look
a t c e r t a i n transformations o f the square
[:]
and
[
.
maps
r e s p e c t i v e l y , onto
As shown in Figure 5-3,
ORST, determined by the basis vectors
the f u n c t i o n
Figure 5-3.
Reflection in the o r i g i n .
Another transfontation that is readily visualized is the r e f l e c t i o n in the
x
axis,
For t h i s mapping, the point
(x,y)
goes into the p o i n t
see Figure 5-4.
Figure 5 4 .
That
is,
Reflection in the
x
the map is given by
Using a matrix, we may rewrite t h i s r e s u l t in the form
[see.
5-11
axis.
(x,--y) ;
as
is e a s i l y v e r i f i e d .
the square
NOW this
unchanged; thus the vector
(1,O)
[]
y
i s
ORST,
leaves the point
mapped o n t o i t s e l f
.
The v e c t o r
is mapped o n t o
Reflection in the
1 8 0 ~ in space about the
axis, or a rotation of
y
a x i s , can be expressed similarly:
as shown in Figure S 5 .
F i g u r e 5-5.
Rotation of
about the axes.
189'
Casual observation (see Figures 5-5 and 5-6)
a
180'
the
r o t a t i o n about the
(x,y)
onto
axis and
90'
pLane are equivalent; they are n o t .
leaves t h e point
(0,l)
y
(0,l)
-10).
may l e a d you to assume t h a t
rotation about the origin in
The f i r s t transformation
unchanged, whereas the second transformation maps
As a vector function, the
90'
rotation with respect t o
the origin is expressed by
Figure S 6 .
Rotation of
The transformations of Figures 5-2
size n o r the shape of the square.
does a l t e r size.
90
0
about the origin.
through 5 4 have altered neither the
The "stretching" function of Figure 5-1,
A "shear" that moves each p o i n t parallel t o the
x
axis
through a distance equel to twice the ordinate o f the p o i n t a l t e r s shape.
Con-
sider the transformation
which maps the basis vectors onto
The result is
Figure 5-7.
a sheering t h a t transforms the
square into a parallelogrflm; see
What does the s t r e t c h i n g do t o shape?
f sec. 5-11
The shear to size?
Figure 5-7.
Shearing.
Another type of transformation involves a displacement or "translation"
in the direction of a fixed vector.
V
The mapping
+ V + U,
where
U =
I:1
*
can be written in the form
One way of visualizing this function is to regard i t as translating the plane in
the direction of the vector
U
through a distance equal to t h e length of
U.
Tran~formationof two d i f f e r e n t t y p e s can be combFned in one function.
For
i n s tance , the mapping
V
1
+?(V
+ U),
where
U =
involves a translation and then a compression.
When the function is expressed
in the form
ue recognize more easily that every point P is mapped onto the midpoint
the line segment joining P to the point: (3,Z); see Figure 5 8 .
[sea. 5-11
of
Figure 5-8. The transformation V
Under t h i s mapping, the square
1
+-i.(V
+ U) .
will likewise be translated
1
toward the p o i n t (3,2) and then compressed by a factor
, as shown in
2
Figure 5-9. This f i g u r e enables us to see that the points 0, R, S, and T
ORST
-
are mapped onto midpoints of l f nes connecting these p o i n t s t o ( 3 , 2 ) .
Figure 5 - 9 .
Translation and compression.
A l l the vector functions d i s c u s s e d above map d i s t i n c t points of the plane
onto d i s t i n c t p o i n t s .
This i s not always the c a s e ; we can c e r t a i n l y produce
functions that do not have t h i s property.
[sec.
Thug, the f u n c t i o n
~ l l
On the other hand, the trans-
maps every poinr of the plane onto the o r i g i n .
formation
maps the poinr
component as
the p o i n t
(x,y)
onto the point o f the
axis that has the same f i r s t
x
For example, every p o i n t o f the line
V
(3,O).
Since the image o f each point
a p e r p e n d i c u l a r Line £rum P
carried or projected on the
t o the
x
x
= 3
is mapped onto
can be located by drawing
P
axis, we may think of P as being
x
axis by a l i n e perpendicular t o this axis.
Con-
sequently, t h i s mapping may be described a6 a p e r p e n d i c u l a r or orthogonal
projection o f the plane on the
map
H
o n t o subspaces
of
axis.
x
You notice that these Last rw functions
H.
Since we have met examplea o f transformations t h a t map d i s t i n c t p o i n t s onto
distinct points and have also seen transformations under which distinct points
may have the same image, i t i s u s e f u l to define a new term t o d i s t i n g u i s h be-
tueen these two kinds
D e f i n i t i o n 5-1.
of
vector f u n c t i o n s .
A transformation frum the s e t
H onto the s e t
H
is one-
tcrone provided that the images of d i s t i n c t vectors are also d i s t i n c r v e c t o r s .
f
Thus, if
the image of
V
is a f u n c t i o n from
H onto H
under the transformation
formulated symbolically as follows:
f,
and if we write
then Definition 5--1
The function
tion of H provided that, for vectors
V
and
U
f
Is a on-to-one
in H,
imp lies
Exercises 5-1
1.
Find the image of the vector
V
under the mapping
f(V)
for
can be
transforma-
V
(1)
AV.
The study of the solution of systems of Linear equations thus leads t o the
consideration of the s p e c i a l class of transformations on
in the form (11, where
A
is any
2 x 2
H
that are expressible
matrix with real entries.
These matrix
transformations constitute a very important class of mappings, having extensive
applications in mathematics, statistics, phyefcs, operations research, and
enatneering.
h important properry of matrix transformations Is
chat: they are linear
mappings ; that is, they preserve vector sums and the products o f vectors with
real numbers.
L e t us formulate these ideas explicitly.
Definition +2.
H
is a functfon
U
in H, we have
f
from H
H such that
into
(a)
for every p a i r of vectors
(b)
f o r every real number
Theorem 5-1.
Proof.
where
and
A linear transformation on
U,
A
Let
r
V
and
and every vector
V
in H,
we
have
Every matrix transformation is Linear.
f
be rhe transformation
is any real matrix of order 2 .
We must show that for any vectors
V
we have
A(V
+ U)
+ AU;
= AV
further, we must show that for any vector
V
and any real number
r we have
19
But these equali t i e g hold in virtue o f parts I11 (a) and 111 ( £ 1 o f Theorem 4-2
(see page 1 3 4 ) .
The l i n e a r i t y p r o p e r t y of matrix transformations can be used t o derive the
following result concerning transformations of the subspaces of
A matrix
Theorem 5-2.
F'
maps every subspace
A
F
of
H.
H onto a subspace
H.
of
Ft
Let
Proof.
F1
To prove that
denote the s e t of vectore
i s a subspace of
we must show that the following state-
H,
ments are true:
For any pair of vectors
(a)
in
F'
PI
If
and
Q
F',
the sum
P' + Q '
is
P'
F1 and any real number r,
in
rP'
is in
.
Q'
and
in
F1.
For any v e c t o r
(b)
PI, Q'
are in
F',
then they must be the images of vectors
P
F; that is,
in
P'
=
AP,
Q' = AQ.
It follows t h a t
P'
end
P
PI
+Q
f
Thus,
rP'
= AP + A Q = A ( P 4- Q),
is the image of the vector
Q'
is in
and hence
+ Q'
F?)
rP'
Hence,
(P'
f
i s the image of
Q') E F'.
rP.
+Q
in
F.
(Can
you t e l l why
Similarly,
But
is t h e image o f a vector in
Corollary 5-2-1.
P
rP
F;
f
F because
therefore,
Every matrix maps the plane
Isec. 5-21
rP'
F
is a subspace.
E
F'.
H o n t o a subspace of H,
192
either the origin, or a straight Line through the origin, or H
itself.
For example, t o determine the subspaces onto which
H
(b)
itself,
we proceed as f o l l o w s .
For (a),
the vectors of
F
are o f the form
Hence,
F is mapped onto F',
Thus,
In other words,
A
[t]
the s e t of vector8 collinear with
;
t h t is,
maps the line passing through the o r i g i n w i t h slope -3
onto
the lfne through the origin with slope 1/2.
A8
we
regards (b), we note that for any vector
have
Since
2x
+y
assumes a11 real values
numbers, it folloca that
H
as x
and
fs also mapped onto
entire plane onto the line
[sec. 5-21
y
run over the set of real
F 1 ; that is,
A
maps the
Exercises L 2
I.
Let
A =
[i
:].
For each of the following values of the vector
V,
determine:
( i ) the v e c t o r into which A maps
(ii)
2.
the line onto which
A
V,
maps the line containing V,
A certain matrix maps
I:[
into
[:] ,
and
[ i]
into
[r] .
Using t h i s i n f o r m a t i o n , determine the vector i n t o which the matrix maps
each of the following:
3.
Consider the f o l l o w i n g subspaces of
F3
=[:I
: Y E -
2x
I
,
H:
F4=H
itself.
194
Determine the subepaces onto which
F1, F 2 , F 3 , and
F4
are mapped by each
of the following matrices:
a
5.
=
[ ]
,
AV
(a)
Calculate
(b)
Find the vector
(8)
3
- [ '1
-2O
3
'
(GI
*g5
(dl
BA-
for
V
for which
Determine which of the following transformations of
H are linear, and
j u s t i f y your answer:
(e)
,V
71 (V
f
U), where
U =
6.
Prove t h a t the matrix A maps the plane onto the origin if and only if
7.
Prove t h a t the matrix
only if
8.
Prove that
A
maps every vector of the plane onto itself if and
w5
maps the line
y = 0
IS any point
onto itself.
of
that line mapped onto
itself by t h i s matrix?
9.
(a)
Show that each of the matrices
H onto the x axis.
maps
(b)
Determine the s e t of aLL matrices that: map
(Hint:
H onto the x
You muat determine all p o s s i b l e matrices
corresponding t o each
V G H
A
axis,
such t h a t
there is a real number
r
for
which
[;I
In part cular, (1) mus
Y
[
andby
hold for s u i t a b l e
Determine the s e t of a l l matrices that map
11.
(a)
f o r all
(b)
V
is replaced
h )
LO.
Determine the matrix
r when
H onto the
y
axis.
such t h a t
A
V.
The mapping
multipliee the lengths of all vectors without changing their directions.
It amount6 t o a change of scale.
scale factor or scalar.
The number
Find the matrix
A
a
i~
accordingly called a
that yields only a change o f
scale:
12.
Prove that for every matrix
A
the s e t
[see. ?2]
F
o f all vectors
U
f o r which
196 .
13.
This subspace is called the kernel of the mapping.
H.
is a subspace of
Prove that a transformation
r
5-3.
and
U
into itself is linear if and only i f
of
H
and every pair of real numbers
s.
Linear Trans formations
I n the preceding s e c t i o n ,
H
transformation of
formation of
H
we proved chat every matrix represents a linear
into 8. We now prove the converse:
into H
can be represented by a matrix.
Let
f be a linear transformation of H
Theorem 5-3.
relative to any given basis for
such t h a t , f o r all
Proof.
ing
and
V
for every pair of vectors
of B
f
V
E
H,
into
H.
Then,
there e x i s t s one and only one matrix
A
H,
We prove f i r s t that there cannot be more than one matrix represent-
Suppose that there are t w o matrices
f.
Every linear trans-
A
and
B
such that, for all
V E kt,
AV = f (V)
and
BV = E(V)
.
Then
AV
for each
V.
that
A
A
-B
-B
=
f (V)
- f (V)
Hence,
(A
Thus,
- BV
- B)V =
[i]
for all
maps every vector onto the o r i g i n .
is the zero matrix; therefore,
V t
X.
It follows ( E x e r c i s e 5 - 2 4 )
197
Hence, there is at most one matrix representation of
f.
Next, we show how t o f i n d the matrix representation for the linear transfor-
f.
mation
Let
S1
and
S2
be a pair of noncollinear vectors of
be the respective images of S L and
vector of
and v
2
H,
S2
under the mapping
f.
If
H.
V
k t
is any
it follows from Theorem 4-10 that there exist real numbers
such that
V = vlSl
+ vZS2.
Since
f
vl
is a linear transformation, we
have
Accordingly,
Thus,
It follous that
f
is r e p r e ~ e n t e dby the matrix
when vectors are expressed in terms of their coordinates relatlve t o the basis
S1' S*
You notice that the matrix A is completely determined by the e f f e c t of f
on the pair of noncollinear v e c t o r s used as the basis for H. Thus, once you
know that a given transformation on H is linear, you have a matrix representing the mapping when you have the imagee of the natural basis vectors,
For example, I t can be shown by a geometric argument that the counrerclock-
Isec* 5 3 1
198
0
wise rotation o f the plane through an angle of 30
rransformarion.
This function maps any point
measure of the angle POP'
(Figure 5-11}
Figure >10.
P
about the origin is a linear
onto the p o i n t
PI,
where the
(Figure 5-10). It is easy t o see
i s equal t o 30'
that
A rota tion through
en angle of 30'
Figure 5-1 1.
about the o r i g i n ,
(1,O)
The images of the points
and (0,l) under a r o t a t i o n of 30'
about the o r i g i n .
[;I
is rapped onto
[in:
$1
and
Thus, the matrix representing this rotation is
I..
COS
A =
m0
Note that the f i r s t column of A
the second c o l m n of
A
I.,
c..
[ ; 41
-
s i n 30'
30°
=
--
is the vector onto which
is the image o f
]
a
irmapped;
Ly1
The product or composition of two rransfowations is d e f i n e d just as you
[see.
5.31
define the composition of two real functions o f a real variable.
Definition 5-3.
vector
V
in
H
If
and
f
g
are transformations on
the composition transformations
Lg
and
then for each
H,
are the trans-
gf
formations such t h a t
fg(V)
f(g(V))
a
Thus, to find the image of
g,
apply
f
maps
f.
and then apply
U onto
W,
then
V
gf(V) = g(f(V1).
and
under the transformation
Consequently, if
fg maps
V
maps
g
V
A,
and
and
gf
onto
by
and i f
is a linear transformation represented by the matrix
f
are both linear transformations;
i s represented
U,
.
is a linear transformation represented by the matrix
g
you first
W.
onto
The following theorem is readily proved (Exercise 5-3--7)
Theorem S . If
fg,
is represented by
fg
B,
AB,
then
fg
while
BA.
For example, suppose that in the coordinate plane each position vector is
f i r s t r e f l e c t e d in the
Ln length.
y
axis, and then the resulting vector is doubled
L e t us f i n d a matrix representation o f the resulting linear tratls-
formation on
H.
If
g
i s the mapping t h a t transforms each v e c t o r into its
reflection in the vertical axis, then we have
I Y I
I-
1-1
I
nl I Y l
If f maps each vector into twice the vector, then
Accordingly, the m a t r i x representing
fg
is
we have
gf
200
Exercises 5-3
1.
Show that each o f the mappings in Exerci a e 5-1-3
is linear, by determining
matrices representing the mappings.
2.
Conaider the linear transformations,
of
p:
reflection in the horizontal axis,
q:
hotizontal p r o j e c t i o n on the l i n e
r:
rotation counterclockwise through 90
a:
shear moving each p o i n t vertically through a distance equal to
the abscissa of the p o i n t ,
y =
0
-x
(Exercise H - 5 e ) ,
,
H into H. In each of the following, determine the matrix represent-
ing the g i v e n transformation:
3.
Let
f
be the rotation of the plane counterclockwise through 45O about
the o r i g i n , and l e t
P
g
be the rotation clockwise through .
'
0
3
matrix representing the rotation cmterc~oclruisethrough 15'
Determine
about the
origin.
4.
(a)
(b)
Prove that every linear transformation mape the origin onto i t s e l f .
Prove that every linear transformation maps every subspace o f
a subspace of
5.
f and g on H,
For every two linear transformations
be the transformation such that, for each
(f
+ g)(V)
= f(V)
Mithou t using matrices, prove that
6.
For each linear transformation
af
H
onto
+g
to
H.
f
f
to be the transformation such that
H
f
V E H,
+ g(V1.
+g
on
define
is a linear trans formation on
and each real number
a,
define
Ii.
201
Without using matrices, prove that
is a linear transformation on
af
H.
7 . Prove Theorem 5-4.
Without using matrices, prove each of the following:
8.
+ h)
(a>
f(g
(b)
(f
(c>
f(ae)
where
f,
+ g)h
= f g -4-
+ gh,
= fh
= a(fg),
and
g,
fh,
h
are any linear transformations on
and
H
a
i s
any real number.
5-4.
One-to-one
Linear Transformations
The reflection of the plane in the
axis clearly maps d i s t i n c t p o i n t s
x
onto distinct points; thus, the reflection is a one-to-one
linear transformation
on H. Moreover, the reflection maps any pair of noncollinear
pair of noncollinear vectors.
It is easy t o show t h a t this property is common
H into itself.
linear transformations o f
to all one-to-one
Theorem 5-5, Every o n e t o a n e linear transformation on
v e c t o r s onto
Proof.
onto a
vectors
H maps noncollinear
noncollinear v e c t o r s .
Let
S1
and
S2
f(S )
1
be a pair of noncollinear vectors and let:
=
T1
be their images under the one-to-one
one, we know that
therefore, that
T1
TI
and
T,
t
and
f ( S . 1 = T2
L
l i n e a r mapping
f.
Since
are not both the zero v e c t o r .
i s not the zero vector.
To show that
f
is one-to-
W
e may suppose,
TI and T2
are
not c o l l i n e a r , we shall demonstrate that the assumption t h a t they are collinear
leads to a c o n t r a d i c t i o n .
If TI and
that
f
T2
= r T.I
T2 are collinear, then there exists
Now, c o n s i d e r the image under
is linear, we have
f
a
real number
of the v e c t o r
r
r S1.
such
Since
202
~ h u s ,each of the vectors
S2
is mapped onto
T2. Since
f
is one-
i t follows rhar
t-ne,
and therefore that
the fact that
and
r S1 and
S1 and
S1 and
S2
are collinear v e c t o r s .
are not collinear.
S2
T2 are collinear must be false.
But this contradicts
Hence, the assumption t h a t
T~
f must m a p noncollinear
Consequently,
v e c t o r s onto noncollinear vectors.
Corollary 5-5-1.
H
maps
is
Proof.
H
The subspace onto which a one-to-one
linear transformation
itself.
Since the subspace contains a pair of noncollinear vectors, the
corollary follows by use of Theorems 4-9 and 4-10.
The link between one-to-one
transformations on
H and second-order
matrices having inverses i s given in the next theorem.
Theorem 5-6.
Then
A,
H
Let
f
be a linear transformation represented by the matrix
f
is one-to-one
if and o n l y if A
Proof.
Suppose that
A
having the same image under
has an inverse,
f.
f(sl) =
Thus,
Hence,
and
has
an inverse.
Let
S1 and
Now,
AS1
and
f ( S 2 ) = AS2,
S2
be vectors in
Thus,
f
must be a o n e t o - o n e trans f o r m a t i o n .
On the other hand, suppose that
follows that every vector in
l a r , there are vectors
W
H
and
f
i s
From Theorem 5-5, it
one-to-one.
i s the image of some vector in
U
H.
In p a r t i c u -
such that
and
Accordingly, the matrix having for its f i r s t column the vector
its aecond column the vector
U,
is the inverse
of
W,
and for
A.
A linear transformation represented by the matrix
Corollary 5-6-1.
A
is
o n e - - t ~ n e if and only i f
The theory of systems of tw linear equation8 in two variables can now be
s t u d i e d geometrical t y
.
Writing the sys tern
V
t h a t are mapped by the matrix
i n the form
where
we seek the vectors
Isec.
3-4.3
A
onto the vector
U.
2@+
If 6(A) # 0 ,
onto
H.
we now know t h a t
Therefore,
A
A
represents a one-tc-one
- has
Thus, the s y s t e m (1) - or, equivalently, (2)
If
F(A) = 0,
maps
A
a
and
Hence,
Xf
V =
A
U.
the coltrmns of
A
must
has the Ef rs t of these forms, then
A
In the o t h e r two cases,
H onto the o r i g i n .
namely,
m u s t have one of the forms
A
are z e r o .
b
U,
H
-1
exactly one s o l u t i o n .
then, in v i r t u e of C o r o l l a r y 4-7-1,
be collinear v e c t o r s .
where n o t both
onto
V
maps exactly one vector
mapping of
[;I
line of vectors collinear with the vector
.
A
maps
(See Exercise
H o n t o the
54-7, b e l o w . )
With these results in mind, you may now complete the discussion of the s o l u t i o n
o f equation ( 2 )
.
Exercises 5-$
1.
Using Theorem 5-6
in Exercise 5-1-3
2.
or its corollary, determine which of the transformations
are one-to-one.
Show thar a linear rransformation is o n e - t o o n e if and only if the kernel
o f the mapping c o n s i s t s only of t h e zero vector.
3.
(a)
Show that i f
(See Exercise 5-2-12
is a o n e - t m n e linear transformation on
f
there e x i s t s a linear transformation
g
such that, for a l l
H,
.)
then
V c H,
and
The transformation
(b)
Prove that i f
fg
i a called the inverse o f
Show that the transformation
transformation on
4.
g
-1
g = f
f
and is usually written
in part: (a) i s a one-tcrone
H.
f and
is also a one-to-one
g
are
onetrrone
linear transformations of
transformation of
H.
H,
then
205
5.
Prove that the s e t of one-t-ne
H
linear transformations on
i s a group
relative to the operation of composition of transformations.
6.
Show t h a t i f
are linear transformations o f H
f and g
i s a o n e t m n e transformation, then both
f
and
g
A
maps
such that
are onetc-one
fg
trans-
forraations,
7.
(a)
Show that if
&(A) = 0,
then the matrix
H
onto a point
( t h e o r i g i n ) or onto a line.
Shov that i f
A
i s the zero matrix and
then every vector
V
of
(b)
is
the zero vector,
is a s o l u t i o n of the equation AV = U.
H
Shov that if 6(A) = 0, but
(c)
U
A
is not the zero matrix, then the
A
is
solution s e t o f the equation
is a s e t of collinear vectors.
Show that if 6(A) = 0, but
(d)
not
the zero matrix,and
U
is
not the zero v e c t o r , then the solution set of the equation
either is empty or consists
where
V1
and
V2
of
are fixed vectors such that
AV1 = U
8.
Show that if the equation
U,
55.
then
A
all vectors of the form
AV = U
and
AV2 =
[:I
has more than one s o l u t i o n for any given
does not have an inverse.
Characteristic Values and Characreristic Vectora
If we think of a mapping as "carryingt' p o i n t s of the plane onto other points
o f the plane, we might ask, through curiosity, i f there are
cases
in which the
206
image p o i n t under a mapping
i s
the same as t h e p o i n t itaelf.
Such 'fixedf
p o i n t s , or vectors, are o f great importance in mathematical a n a l y s i s .
Let us look at an example.
The r e f l e c t i o n with respect t o the
axis,
x
that i s ,
has the property of mapping each vector on the
axis onto i t s e l f ; thus,
x
each of these vectors is fixed under the transformation.
Definition 5 4 .
If a transformation of H
into itself maps a g i v e n vector
onto Ftaelf, then that: vector is a fixed vector for the transformation,
More generally, we are interested in any vector that is mapped into a
m u l t i p l e of i t s e l f ; that i s , we seek a vector
V
E
H and a number c E R
such
rha r
Since the equation is automatically s a t i s f i e d by the zero vector regardless of
the value
c,
t h i s v e c t o r i s ruled o u t .
The number
the vector
V
c
is called a characteristic value (or eigenvalue)
a characteris t i c vector o f A.
A, and
of
These notions are fundamental in
atomic physics since the energy levels of atoms and molecules turn out to be
given by the eigenvalues of certain matrices.
A l s o the analysis o f f l u t t e r
and vibration phenomena, the stability analysis of an airplane, and many other
physical problems require finding the characteristic values and vectors of
matrices.
In Section 5 1 , we
carried the plane
saw t h a t the mapping
H onto the line
y = x/2.
I f we consider the s e t
F
207
under t h i s same mapping, we see that
collinear with
F.
F
i s mapped onto
F'
, the
s e t of vectors
Note that
and hence t h a t 5 is a characteristic value associated with
a characteristic vector for any
D e f i n i t i o n 5-5.
t c R,
t
#
A,
and
12:1
L
0.
4
Is
Each nonzero vector s a t i s f y i n g the equation
i s called a characteristic vector, corresponding to the characteristic value
(or characteristic root)
c
of
A.
Note that, as remarked above, the t r i v i a l s o l u t i o n ,
is not considered a c h a r a c t e r i s t i c vector,
[:I
of the equation
Because of the importance of characteristic values i n pure and a p p l i e d
mathematics, we need a method for f i n d i n g them.
and real numbers
c
we seek nonzero vectors
such t h a t
If I is the i d e n t i t y matrix of order 2, then (1) can be written
If
we l e t
equation ( 2 ) becomes
as
V
v
We know t h a t there is a nonzero vector
s a t i s f y i n g an equation of the form
BV = 0
if and only if
Hence equation ( 2 ) has a s o l u t i o n o t h e r than the zero vector if and o n l y if
is chosen in such
a
c
way as ro s a t i s f y the e q u a t i o n
Rearranged, this equation becomes
which is called the characteristic equation of the matrix
quadratic equation is solved f o r
the corresponding vectors
c,
equation (1) can readily be found, as
Example.
Once t h i s
V
satisfying
illustrated in the following example.
Determine the fixed lines under the mapping
We must solve the matrix equation
Since
6(A
A.
- cI) = ( 2
[see*
- c)(l - c ) ,
5-51
the characteristic equation is
the roots o f which are
1
and
2.
For
c = 1,
equation ( 3 ) becomes
which is equivalent t o the system
Thus,
A
maps the l i n e
is mapped onto i t s e l f .
is invariant:
if V
x
+
3y = 0
onto i t s e l f ; t h a t i s , the s e t
Actually, since
i s a c b r a c t e r i s t i c vector,
then
f(V)
For
c = 2,
Hence,
c = 1,
equation ( 3 ) becomes
-
V.
F
each vector of t h i s s u b s p a c e
f(V)
i t s image, and
c = 1,
y = 0
maps the line
onto i t s e l f ; the s e t of vectors
F
is closed under the transformation.
The characteristtc equation associated with the matrix
This equation expresses a real-number function.
For a matrix function, the
corresponding equation i s
where
order 2.
If we substitute
we f i n d that
any
and 0 IB
the zero matrix of
in this matrix equation,
I is the i d e n t i t y matrix o f order
2 x 2
A
is
a
A
2
root of i t s characteristic equation, This La true f o r
matrix.
Theorem F7. The m t r i x
A
is a s o l u t i o n o f its characteristic equation
The proof is left as an exercise.
Theorem 5-7 is the case n = 2
o f a famoua theorem called the Cayley-
Hamilton Theorem, vhich s t a t e s that an analogous result holds for matrices of
any order
n.
Exercises 5-5
1.
Determine the characteristic roots and vectors o f each of the follouing
matrices:
2.
Prove that zero is a characteristic root of a matrix
6(A)
3.
= 0.
Show that a
linear transformation
f
is one-t-ne
if and only i f zero i s
not a characteristic root of the matrix representing
4.
if and only i f
A
f
.
Determine the invariant subspaces (fixed Lines) of the mapping given by
Show t h a t these l i n e s are m ~ c u a l l yp$rpendicular.
is a solution of i t s characteristic
22
(matrix) equation
6.
Show that
but =hat 2
is
[i]
an
invariant vector of the transformation
is not invariant under this mapping.
23.2
7.
Show that
maps every line through che o r i g i n onto itself if and only if
A
A
8.
for
r # O .
Let
d = (al1 - a Z 2 )
L
real numbers.
+4
=
[: or]
a
where
12a21'
Show rhat the number
a l l ' a12' =21' and aZ2 are any
o f d i s t i n c t real characteristic roots
of the matrix
9.
Find a nonzero matrix t h a t leaves no Line through the o r i g i n f i x e d .
linear transformation that maps exactly one line
10. Determine a one-te-one
through the origin onto i t s e l f .
11,
Show rhat every matrix
roots i f
12.
B
o f the form
# 0.
Show that the matrix
A
[::I
and i t s transpose
has two d i s t i n c t characteris tic
have the same characteristic
A~
roots.
5-6.
Rotations and Reflections
Since length is an important property in Euclidean geometry, we shall look
for the linear transformations of the plane that l e a v e unchanged the length
I I VI I
of every vector
V.
Examples of such transformetions are the following:
(a)
the reflection of the plane in the
(b)
a r o t a t i o n of the p l a n e through any g i v e n angle about the o r i g i n ,
(c)
a reflection
x
axis,
in t h e x axis followed by a ra t a t i o n
about the o r i g i n .
I
2u
A c t u a l l y , we can show that any linear transformation that preserves the lengths
of a l l v e c t o r s i s e q u i v a l e n t t o one of these three.
The following theorem w i l l
be very u s e f u l i n proving t h a t r e s u l t .
Theorem 5-8.
A linear transformation of
H
that leaves unchanged the
l e n g t h of every vector also leaves unchanged (a) the inner product o f every pair
of vectors and ( b ) the magnitude o f the angle between every pair of v e c t o r s .
Proof.
Let
V
and
U
be a pair of vectors in
be t h e i r respective imegea under the transformation.
4-Hi,
we
and l e t
H
Vt
and
U'
In virtue of Exercise
have
and
Since the transformation is linear, f o r the image of
Consequently, (2) can be written as
II(V
+u)'112
= IIV'II
2
+ 2Vi
+U
V
.
U'
+
we have
IIV'II
2
.
But the transformation preserves the Length o f each v e c t o r ; thus, we obtain
IIV'II = IIVII,
IIU'II = IIUII,
and
II(V f U)'II
-
IIV
+ Utl.
Making these s u b s t i t u t i o n s i n equation (3), ue g e t
Comparing equations (1) and ( 4 ) , you see that we must have
t h a t i s , t h e transformation preserves the value of the inner p r d u c t .
[see. 563
(3)
a4
Since the magnitude of the angle between
V
U
and
can be expressed in
terms of i n n e r products ITheorem 4-51, it f o l l o w s that the transformation also
preserves that magnitude.
If a linear trans formarion preserves the length of every
Corollary 5-8-1.
vector, 'then it maps orthogonal vectors onto orthogonal v e c t o r s .
By the d e f i n i t i o n o f orthogonality, t h i s simply means t h a t the geometric
vectors are mutually perpendicular.
X t is very easy to show the transformations we are considering a l s o pre-
serve t h e d i s t a n c e between every pair of points in the plane.
Ke state this
property formally i n the next theorem, the proof of which is left as an exercise.
A linear transformation that preserves the length of e v e r y
Theorem 5-9,
vector leaves unchanged the distance between every p a i r of p ~ i n t sin the plane;
V'
that i s , if
U'
and
are the respective images of the v e c t o r s
V
and
U,
then
Let us now f i n d a matrix representing any given l i n e a r lengttr-preserving
t r a n s f o m a t i u n of
All we need to f i n d are the images of the vectors
H.
under such e transformation.
If
Si
that both
and
Si and
(Why is this s o ? )
are the respective images of
S;
Si
are o f Length
L
SL
and that they
and
S2,
then we know
are orthogonal to each
other.
Suppose t h a t
the
x
S;
axis (Figure 5-12).
We know that:
S;
(alpha) with the positive half o f
forms the angle
Since the length of
is perpendicular to
S;
.
S;
equals
Hence, there are
1,
we have
two o p p o s i t e
Figure 5-12
.
A length-preserving trans formation.
possibilities for the direction of
S;,
because the angle
makes with the positive.half of the
x
axis may be either
In the firs t case (5) , we have
I n the second case (61, we have
n
[sec.
561
s i n Cr
P
(beta) that
Si
236
AccordingLy, any linear transformation
t h a t leaves the length of each
f
vector unchanged must be repreeented by a matrix having either the form
cos
a -sin
[r
I
or the form
In
the f i r s t instance (71, the transformation
vectors
SL and
o f the entire plane
the v e c t o r
r = IIVII;
f
is a rotation
To v e r i f y t h i s observation, we write
through that angle.
H
in terms o f its angle a f inclination
V
and the Length
Forming
and we suspect that
through an angle
S2
simply rotates the basis
f
(theta) t o the
9
x
axis
that is, we write
from equations ( 7 ) and (9), we o b t a i n
AV
[
AV =
I
r(cos 9 cos
- s i n 8 sin a)
r(sfn 8 cos cX + cos 8 sin a)
'
From the formulas of trigonometry,
w e see
+ a) =
( 8 + a) =
cos ( 0
cos 8 cos
a - sin 8
sin
s i n 8 cos
(Y
+ cos
Q sin
a,
a,
that
=
[
is the vector o f length
r
A.
Thus,
AV
axis.
We have proved that the matrix
the angle
r cos (9
r s i n (8
+ a)
+ a)
a t an angle
A
8
+ Cr
to the horizontal
represents a rotation of
fl
through
(T.
But suppose
f
i s
represented b y the matrix
B
in equation (8) above.
This transformation differs from the one represented by
S;
sin
is reflected across the line of the vector
S;.
A
in that the vector
Consequently, you may
suspect t h a t rhFs transformation amounts to a r e f l e c t i o n of the plane in the
[sec.
2-61
217
axis followed by a rotation through the angle
x
reflection in the
you may
x
therefore
Since you know that the
LY.
axis is represented by the matrix
expect that
We leave t h i s verification as an exercise.
Exercises 5 6
1.
Obtain the matrices that rotate
Write
through.the following angles:
(a)
180°,
(£1
(b)
45O,
(g)
(c)
30°,
(h)
(dl
boo,
i
(e)
2.
H
out
270°,
90°,
-120°,
360°,
-13s0,
(j)
LSO'.
the matrices that represent the transformation conaieting of a
r e f l e c t i o n in the
x
axis
followed by the rotations of Exercise 1.
3.
Verify Equatian (lo), above.
4.
A linear transformation of H
t h a t preserves the Length of every vector is
called an orthogonal transformation, and the matrix representing the' transformation is called an orthogonal matrix.
Prove that the transpose of an
orthogonal m a t r i x is orthogonal.
5.
Show that the inverse o f an orthogonal matrix is an orthogonal matrix.
6.
Show that the product of two orthogonal matrices i s orthogonal.
7.
(a)
Show that a translation of
In the direction of the vector
and through a distance equal to the length o f
[sac. 5-61
U
is g i v e n by the mapping
(b)
Show that this mapping does not preserve the l e n g t h of every vector,
b u t t h a t it does preserve t h e d i s t a n c e between every pair o f p o i n t s in the
plane.
8.
(c)
Determine whether or n o t t h i s mapping is linear.
Let
Ra
and
respectively.
through
9.
B
denote rotations o f
(a)
H
through t h e anglea
Prove that a rotation through
0 amounts to a rotation through CI
Note t h a t the matrix
number.
0 .
R
A
Q
+ @;
a and
followed by a r o t a t i o n
that
i s , show
that
of Equation (7) is a representation of a complex
What does the result o f Exercise 8 imply f o r complex numbers?
Find a matrix that represents a reflection across the line of the
vector
(b)
b,
Show that t h e matrix
B
of equation
flection across the Line of some v e c t o r .
(a),
above, represents a re-
Appendix
RESEARCH EXERC X S E S
The e x e r c i s e s i n t h i s Appendix are essentially "research-type" problems
d e s i g n e d to exhibit aspects of theory and practice in macrix algebra t h a t c o u l d
n o t be included in the t e x t .
They are especially suited as i n d i v i d u a l assign-
ments for those students who are p r o s p e c t i v e majors i n the theoretical and
p r a c t i c a l aspects o f the scientific d i s c i p l i n e s , and for students who would like
AlternativeLy, small groups of s t u d e n t s
to test their mathematical powers.
might join forces in working them.
1.
Quaternions.
was invented by the
The algebraic system that is explored in this e x e r c i s e
Irish mathematician and physicist, William Rowan Hamilton,
who p u b l i s h e d his first paper on the subject in 1835.
It was nor until 1858
that Arthur Cayley, an English mathematician and lawyer, p u b l i s h e d the f i r s t
research paper on matrices, though the nane matrix had p r e v i o u s l y been a p p l i e d
by James Joseph S y l v e s t e r in 1850 t o rectangular arrays of numbers.
Hamilton's
s y s t e m o f quaternions i s
Since
a c t u a l l y an a l g e b r a o f matrices, i t is more
easily presented i n this guise than in the form in which i t was first developed.
In the present exercise, we shall consider the algebra o f
w i t h complex numbers as e n t r i e s .
and inversion remain the same.
and we denote by
where
z, w, zl,
K
and
K
matrices
The definitions of a d d i t i o n , m u l t i p l i c a t i o n ,
We use C
for the set of all complex numbers
the s e t o f a11 matrices
w
L
are elemerlts o f
having real entries, the element
of
2 x 2
has an inverse if and only if
C. As i s the case w i t h matrices
220
and then we have
Since
1
is a complex number, the unit matrix is s t i l l
If
z
a
x 4- iy,
then we write
and call this dumber the complex conjugate of
A quaternion i~ m element
q
[G q] ,
We denote by
(a)
Q
6 ( q ) = x2
Hence conclude that, since
q =
or simply the conjugate of
of the particular form
r t C
and
w E C.
the s e t o f all quaternions.
Ohow that
and only i f
of K
z,
+ y 2 + uZ + v2
x, y , u,
and
v
if
r
-
x
+ iy
are real numbers,
and
w = u
6(q) = 0
if
2.
Show that if q E Q, then q has an inveree if and only if q
and exhibit the form of q-'
if it exists.
(b)
Four elements o f
names :
+ iv.
Q
+ 0- ,
are of particular importance and we g i v e them special
z.
(c)
where
Show t h a t if
z
5
x f iy
and
w ='u
+ iv,
then
q =X
+ yIU + u v * v w .
(d)
Prove the following i d e n t i t i e s involving
I, U,
V
and
W:
v'PV2iJi-f
and
UV=W=-W,
and
VW=U=-WV,
(el Use the preceding two exercises t o show
then q + r , q - r, and q r are all elements of
W=V--uw.
that if
q
E Q
Q.
The conjugate of the element
q =
and the
norm and
[
1,
where r = x
+ iy,
v
-
u
+ lv,
trace are given respectively by
and
(f)
Show that i f
q f Q,
and i f
q
is invertible, then
and
r
E Q,
222
9, and if q-l
From t h i s c ~ n lcu d e that if q -Z
(g)
Show that each q
(h)
Show that i f
6
q
-1
r 9.
s a t i s f i e s the quadratic equation
Q
q E Q,
cnis t a , then
then
Note t h a t t h i s may be proved by using the result t h a t i f
then
and then ueing the results given in ( d ) .
( i ) Show t h a t i f q E Q
and
r e Q,
then
tqrl = Iql I r l
and
The geometry o f quaterniona constitutes a very i n t e r e s t i n g subject.
requires the representation of a quaternion
as a p o i n t w i t h coordinates
(a,
b, e, d)
in four4imenaional spaces.
It
The
subset of elements,
QL
{q: 9
E
Q
and
Iql = I),
is a group and is represented geometrically as the hyperephere w i t h equation
Nonassociative Algebras
2.
The algebra o f matrices (we r e s t r i c t our attention in t h i s exercise to the
set
M
of
cation.
2 x 2
matrices) has an associative but not a comutative multipli-
"Algebras" with nonassociative multiplication have become increasingly
example, in mathema tical gene t i c s .
important in recent years-for
Genetics is
a eubdiacipline o f biology and is concerned w i t h transmission o f hereditary
traits.
Nonassociative "algebras" are important also in the study of quantum
mechanics, a subdiscipline o f p h y s i c s .
We give first:
a simple
algebra (named a f t e r the Norwegian geometer Sophus LLe)
If
A E M
and read this
(a)
and
"A op B,"
"op"
(i)
AoB =
- BOA,
(ii)
AoA =
Q,
(iv)
being an abbreviation f o r operation.
o:
Ao(BoC)
+ Bo(CoA) + Co(AoB)
AoI =
= IoA.
G i v e an example ro show that
and hence that
o
i s not
example of a L i e
we write
Prove the following properties of
liii)
(b)
8 E bf,
.
=
Ao(BoC)
2,
and
(AoB)oC
are d i f f e r e n t
an associative operation,
D e s p i t e these strange properties,
o
behaves nicely relative to ordinary
metrix addition.
(c)
Show that
o
dietributes over addition:
(d)
Show that
o
behaves nicely r e l a t i v e to multiplication by a number.
and
224
It
will be recalled that
and is defined as the
element
A'
is termed the multiplicative inverse of
B
A
satisfying the r e l a t i o n s h i p s
AB =
-
I
BA.
But i t m u s t elso be recalled thar t h i s d e f i n i t i o n was motivated by the fact that
A1 = A =
that is, by the fact t h a t
(e)
IA,
I is a multiplicative unit.
Show rhar there is no
o
unit.
thar
We know, from the foregoing work,
o
i s
neither c o m u t a t i v e n o r
Here is another kind of operation, called Jordan multiplication:
associative.
If A f M and
wedefine
B f M,
AjB =
(AB
+ BA)
2
-
We see at once that
AjB = B j A ,
so that Jordan mulriplicarion is a comutative operation; b u t i t i s n s
associative
(f)
.
Determine all of the properties o f the operation
For example, does
3.
j
j
that you can.
d i s t r i b u t e over a d d i t i o n ?
The Algebra of Subsets
We have seen that there are interestfng algebraically defined subsets of
2 x 2 matrices.
One such subset, f o r example, is the set
M,
the set of all
Z,
which is isomorphic wf th the set of complex numbers.
Much of higher
mathematics i e concerned w i t h the "global structure" of "algebras," and generally
this involves the consideration of eubsets of the "algebras" being s t u d i e d .
t h i s exercise, we shall generally underscore l e t t e r s t o denote subsets of
If
4
and
g are
subsets of
M,
then
In
M.
is the s e t of a l l elements of the form
where
A + B ,
A E
4 and B E -B.
In set-builder notation this may be written
A + B =
-
{A+B
* By an a d d i t i v e subset of
: A s 4
and
B E B].
is meant a subset
M
A
CM
such that
Determine which of the f o l l o w i n g are additive subsets o f
(a)
(i)
M:
(213
(ii>
(11,
(iii)
M,
(iv)
2,
(v)
%,
(vi)
the s e t of a l l
(i)
if
B=
(ii)
(Lii)
A+
0,
-
and if
with
x
(d)
E
2
with
are subsets of
A
-
1,
M,
then
of
M,
then A
- + C-C B-+ C -.
C -B
& and B are addttive subsets
M.
denote the s e t o f a l l column vectors
R
and
Show t h a t i f
5
M whose entries are nonnegative.
V
y c
&(A)
( A + B ) +C,
=
is also an addittve subset of
Let
and
M
!+A,
(B + C-)
Prove t h a t if
(c)
in
t h e set of all elements of
Prove t h a t if A,
(b)
A
R.
v
is
a
fixed element of
V,
then
then
A: A E M
M.
is an a d d i t i v e subset of
set
=
Notice also that i f
If A
- and B are subsets
is the
Av
and
M,
of
AV = 0
then
(-A)"
=
0.
then
of all
AB, A
and
E
B
E
g.
Using set-builder notation, we can w r i t e this in the form
AB =
A subset
A
of
M
IAB: A
6 A
-
and
B
E
01.
is multiplicative if
(e)
Which of the s u b s e t s in p a r t (a) are multiplicative?
(f)
Show t h a t i f
(ii)
and i f
A
-,
B,
-
ACE,
and
are subsets of
C
-
then
M,
then
A
C C BC.
-
A
B of
-,.
M
such t h a t
(g)
Give an example of two s u b s e t s
(h)
Determine which o f the following subsets are multiplicative:
and
(iii)
the s e t of a l l elements o f
M
w i t h negative entries,
(iv)
the s e t o f all elements of
M
f o r which t h e upper left-hand
M
o f t h e form
e n t r y is l e s s than 1,
(v)
the set of all elements of
with
Osx,
Osy,
x + y <-L .
and
The exercises stated above are suggestions as to how this "algebra o f
subsets" works.
There are many other results t h a t come t o mind, but we shall
leave them to you to find.
t 6 R
A- C M?
and
What does
$'
(-l)A- =
Is
mean?
How would you define
Here are some clues:
- -A ?
Wait a minure!
!hat does
-A-
tA_
if
mean?
Does set multiplication distribute over addition, over
union, over i n t e r s e c t i o n ?
Do n o t expect t h a t even your teacher knows the
Few people know all of them and
answer to all of these possible q u e s t i o n s .
fewer s t i l l , of those who know them, remember them.
I f you conjecture that
something is true but the proof o f it escapes you, then try t o construct an
example to show that it: i s false.
If this
d o e s n o t work,
t r y proving i t again,
and so on,
4.
Analysis and Synthesis o f Proofs
This is an exercise i n analysis and s y n t h e s i s , r a k i n g an old proof to
In describing his a c t i v i t i e s ,
pieces and using the p a t t e r n t o make a new proof.
e'mathematician i s l i k e l y to put a t the very top t h a t of creating new results.
Rut "re~ult" in mathematics usually means "theorem and proof."
The mathematician
does n o t by any means lLmit his methods in conjecturing a new theorem:
He
guesses, uses analogies, draws diagrams and figures, sets up physical models,
experiemente, computes; no holds are barred.
Once
he has his conjecture firmly
in m i n d , he is only half through, for he still must construct a p r o o f .
One way
of doing t h i ~is t o analyze proofs of known theorems that are somewhat l i k e the
theorem he is trying to prove and then synthesize a proof o f the new theorem.
Here we ask you t o a p p l y this process of analysis and synthesis of proofs to
the algebra o f matrices.
To accomplish this,
we
shall introduce some new
operatians among matrices by analogy with the old operations.
For simplicity of computation, we shall use only
2 x 2
matrices.
To start w i t h , we introduce new operations in the s e t of real numbers,
If
x
E
R
and
x
/\ y
y E R,
we define
= the
smaller of
x
and
y
(read:
"x
cap y")
"x
cup y").
and
x V y = the larger of
x
and
y
(read:
R.
228
(a)
Show t h a t if
x E R,
y
E
R,
and
z
R,
f
then
(i)
x A y a y A x ,
(ii)
v
x V y = y
x,
/\ y) A z,
( i i i ) x A (y A
2)
= (x
2)
= (xVY)V
(iv)
x
V (Y V
(v)
X
A
(vi)
(vii)
(viii)
2,
X = X,
x V x = x ,
x
A Iy V
Z)
= (X
A y) V (x A z),
x V ( y A z) = (x V y) A
(X
V
2).
Although the foregoing operations may seem a l i t t l e unusual, you will have
no d i f f i c u l t y in proving the above statements.
They are not: d i f f i c u l t to
remember if you n o t i c e the following facts:
The even-umbered
, and conversely.
thar A i s comurative and the t h i r d
and V
interchanging
The first s t a t e s
associative.
r e s u l t s can be obtained from the odd-numbered r e s u l t s by
The f i f t h is new b u t the seventh
s t a t e s that
s t a t e s that
r\
A
is
d i s t r i b u t e s over
v.
To define the matrix operations, l e t us think o f
muLtiplication and
V
A
as the analog of
as the analog of a d d i t i o n and l e t us begin with our new
matrix "mu1 t i p l i c a t i o n . "
We define
This is simply the r o w b y c o l u m n operations, except that
place o f multiplication and
V
is used in place o f addition.
A
is used in
To
see
t h i s more
c l e a r l y , we write
(b)
Write out a proof that i f
A,
B,
and
C
are elements of M ,
then
229
B e sure not t o omit any steps in the proof.
Using
t R i 6 as a
pattern, write out
a proof that
v e r i f y i n g at each s t e p that you have the necessary r e s u l t s from ( a ) t o make the
proof sound.
L i s t all the properties of the tw p a i r s o f operations t h a t you
need, much as a s s o c i a t i v i t y , conrmutativity, and d i s t r i b u t i v i t y .
Using the analogy between
(c)
A
and
(d)
B
of
V
and addition, define
A V
0
for elements
M.
S t a t e and prove, for the new operations, analogs of a l l the rules
you know f o r the operations
of matrix addition and multiplication.
BIBLIOGRAPHY
C. B. Alledoexfer and C. 0. Oakley, "Fundamentals of Freshman Mathematics,"
M c G r a H i l l Book Company, Znc., New York, 1959.
Richard V. Andree, "Selection from Mdern Abstract Algebra," Henry Holt and
Company, 19 58.
R. A. Beaumont and R. W. Ball, "Introduction to Modern Algebra and Matrix
Theory, ' I Rinehart and Company, New York , 19 5 4 .
Garrett Birkhoff and Saunders MacLane, "A Survey of mdern Algebra," The
Mamillan Company, New York, 1959.
R. E. Johnson, "First Course in Abstract Algebra," Prentice-Hall, Inc., New York,
1558.
John G. Kemeny, J. Laurie Snell, and Gerald L. Thompson, " I n t r o d u c t i o n to F i n i t e
Mathematics, " PrenticefIall, Inc , New York, 1957.
.
Lowell J. Paige and J. Dean Swift, "Elements of Linear Algebra," Ginn and
Company, 1961.
M. J . Weiss, "Higher Algebra for the Undergraduate," John Wiley and Sons, Inc
New Y o r k , 1949.
.,
INDEX
Abelian g r o u p , 90
Addition o f matrices, 9
associative law for, 12
commutative law for, 12
A d d i t i o n of v e c t o r s , 147
A d d i t i v e i n v e r s e of a m a t r i x , 14
Additive subset, 225
Algebra, 100, 102
global structure of, 224
nonassociative, 2 2 3
Analysis of p r o o f s , 227
A n a l y s i s , v e c r o r , 175
Angle between v e c t o r s , 1 5 3
An t i c o m u t a t ive matrix, 49
A r e a o f a parallelogram, 1 6 1
Arrow, 136
head, 13b
tail, 136
Associative law, i o r a d d i t i o n , 1 2
Associative law,
for mu1 tiplication, 43-46
Basis, 171
n a t u r a l , 171
Cancellarion law, 37
Cap, 227
Cayley-Hamilton Theorem, 210-,211
Characteristic equation, 2U8
Characteris tic root, 2C6
C h a r a c t e r i s t i c value, 2ij5-2ij7
Characteristic vector, 205-207
Circle, unit, 88
Closure, 53
Collinear vectors, 155
Column matrix, 4
Column of a matrix, 2
Column vector, 4 , 133
Combination, linear, 17U
Comu ra t i v e group, 90
Cammutative law for addition, 12
Complex conjugate, 220
Complex number, 1, 94, 219
Components o f a v e c t o r , 137
Composition o f transformations, 198-149
Compression, 185
Conformable matrices, f o r a d d i t i o n , 10
f o r m u l t i p l i c a t i o n , 27
Conjugate quaternion, 2 2 1
C o n t r a c t i o n f a c t o r , 180
Cosine, d i r e c t i o n , 138
Cosines, law of, 153
Counting number, 1
Cup, 227
Decimal, i n f i n i t e , 1
Dependence, l i n e a r , L 7 L
D e t e r m i n a n t function, 77
Diagonaliza tion method, 131
Difference of matrices, 14
Direction cosines, 138
Direction of a vector, 138
Displacement, 184
Distributive law, 4 1 4 5
Domain of a function, 177
Dot p r o d u c t of vectors, 154
Eigenvalue, 2UC
Electronic brain, 2, 132
Elementary matrices, 124
Elementary row operation, 114, 1 2 4
Embedding of a n algebra, 101;
End p o i n t of v e c t o r , 137
Entry of a matrix, 3
Equality of matrices, 7
Equation, characteristic, 208
Equivalence, r o w , 114
Equivalent systems of linear equations,
105
Equ i v a Len t vectors , 138
Expansion f a c t o r , 179
Factor, contraction, 130
expans i o n , 1 7 9
Field, 5 5
Fixed point, 206
Four-dirnens iona l space, 2 2 2
Free v e c t o r , 175
Function, 177
determinant, 7 7
domain, 177
matrix, l r j 9
range, 177
real, 177
stretching, 183
v e c t o r , 177
Galois, E v a r i s t e , 9 2
Geometric r e p r e s e n t a t i o n of v e c t o r , 140
Global structure of algebras, 224
Group, 85, 90
abelian, Y O
commutative, 90
o f invertible matrices, 85
related to face of clock, 90
Head o f arrow, 1 3 i
Hypersphere, 22.2
I d e n t i t y m a t r i x , f o r a d d i t i o n , 11
for multiplication, 4L
Image, 178
Independence, linear, 2 7 2
Infinite decimal, 1
I n i t i a l p o i n t o f v e c t o r , 137
Inner product of vectors, 152, 154
Integer, 1
Invariant subspace, 211
Invariant v e c t o r , 209
I n v e r s e of a m a t r i x , 62-63, 1 1 3
of order two, 7 5
Inverse o f a number, 54
of a transformation, L O 4
Isomorphism, 9 4 , 100
Jordan multiplication, 224
Kernel, 196
Law o f cosines, 153
L e f t multiplication, 37
Length of a v e c t o r , 137-138
Linear combination, 2 7 0
Linear dependence, 172
Linear e q u a t i o n s , system o f , 103, 119
e q u i v a l e n t , 105
s o l u t i o n o f , 103-105
r e l a t i o n t o matrices, 107
s o l u t i o n by diagonaLization method,
131.
s o l u t i o n by triangularization method,
152
Linear independence, 172
Linear map, 190, 196
Linear trans formation, 190, 196
Located vector, 1 3 6 1 3 7
Map, 178
inverse, 204
k e r n e l , 196
l i n e a r , 190, 196
onetcr-one, 201
onto, 178
Matrices, 1, 3
Matrtx, 1, 3
addition, 9
associative law f o r , 12
c o m u t a t i v e law f o r , 12
c o n f o n n a l b i l i t y for, 10
i d e n t i t y element for, 1 1
a d d i t i v e inverse, 14
an t i c m u tative, 49
column, 4
column o f , 2
conformable far addition, 10
difference, 14
d i v i s i o n , 50--51
elementary, 124
entry o f , 3
equality, 7
i d e n t i t y for a d d i t i o n , 1 1
f o r multiplication, 46
Matrix (continued) ,
inverse, 6 2 4 3 , 113
of order two, 7 5
i n v e r t i b l e , 63
m u l t i p l i c a t i o n , 24, 30, 32
c a n c e l l a t i o n Law for, 37
conformability for, 27
left, 37
r i g h t , 37
multiplicatios by a number, 19-20
negative o f , 14
order o f , 3
orthogonal, 217
product, 26
4
row o f , 2
row,
square, 4
order of, 4
sum, 10
transformation, 189
transpose of, 5
unit, 46
variable, 109
zero, 11
Matrix function, 109
M u l t i p l i c a t i o n . 24, 30, 32
Jordan, 2 2 4
M u l t i p l i c a t i o n o f m a t r i c e s , 24, 30, 32
distributive l a w for, over a d d i t i o n ,
41-45
Pful tiplication o f matrix by number,
19-20
Multiplication o f v e c t o r by number,
144
Natural basis, 171
Negative o f a matrix, 14
Nonassociative a l g e b r a , 223
Norm of a quaternion, 221
N o r m of a vector, 141
Null vector, 139
Number, 1
Number, complex, 1 , 94
conjugate, 220
taunting, 1
integer, L
i n v e r s e , 54
rational, 1
real, 1
One--to-one trans formation, 185
Operation, row, 114, 124
Opposite vectors, 138
Order of a matrix, 3, 4
Orthogonal matrix, 217
Orthogonal projection, 186
Orthogonal transformation, 217
Orthogonal v e c t o r s , 156
parallel vectors, 143, 1 6 3
para1 lelograrn r u l e , 149
perpendicular p r o j e c t i o n , 186
Perpendicular vectors, 153
P i v o t , 132
P o i n t , fixed, 206
Product of transformations, 198
Projection,
orthogonal, 186
perpendicular, 186
m a t e r n i o n , 219-223
conjugate, 221
geometry o f , 222
norm, 2 2 1
trace, 221
Range of a function, 177
Rational number, 1
Real f u n c t i o n , 177
Real number, 1
R e f l e c t i o n , 178, 212
Representation of vector, 140
Right multiplication, 37
Ring, 57-58
with i d e n t i t y e l e m e n t , 60
Rise, 137
Root, characteristic, 206
Row equivalent, 114
ROW matrix, 4
Xow o f a m a t r i x , 2
Row operation, 114, 124
Row v e c t o r , 4
n o t a t i o n , 198, 212
Run, 137
Scalar, 195
Set, 53
c l o s u r e under an operation, 53
element of, 57
Shear, 183
Sigma notarion, 30
Slope of a vector, 138
Space, 1 6 6
fourdimensional, 122
Square m a t r i x , 4
Square root of u n i t matrix, 39
Standard represen tarion, 140
Stretching f u n c t i o n , 183
Subset,
a d d f t i v e , 225
algebra o f , 224
Subspace, 1 6 6 1 6 7
invariant, 211
Sum o f matrices, 10
Synthesis o f proofs, 227
System o f l i n e a r e q u a t i o n s , 103, 119
solution by diagonalization method,
131
Sys tern of l i n e a r e q u a t i o n s (continued)
s o l u t i o n by triangularization method,
132
Tail of v e c t o r , 136
Terminal p o i n t of vector, 137
Trace of a quaternion, 221
Transf o m t i o n ,
composition, 198
geometric, 177-178
inverse, 2r~4
kernel, 146
length-preserving,
l i n e a r , 190, 196
one-to-one, 2 0 1
214-217
matrix, 189
one-to-one, 186
orthogonal , 217
plane, 177-178
product, 198
T r a n s l a t i o n , 184
Transpose of a matrix, 5
Triangularization method, 132
Unit c i r c l e , 88
U n i t matrix, 46
Value,
characteris t i c , 205-207
Variable,
matrix, 139
Vector, 4 , 133
addition, 147
parallelogram rule for, 149
analysis, 1 7 5
angle, 153
basis, 1 7 1
c h a r a c t e r i s t i c , 21~5-2C17
collinear, 1 5 5 , 177
column, 4 , 133
order o f , 133--134
component, 137
d i r e c t i o n , 138
dot product, L54
end p o i n t , 137
e q u i v a l e n t , 138
f r e e , 175
f u n c t i o n , 177
geometric representation, 136
i n i r i a l p o t n t , 137
inner product, 152, 154
i n v a r i a n t , 209
length, 137-138
linear combination, 170
Located, 136-137
m u l t i p l i c a t i o n by a number, 144
natural b a s i s , 1 7 1
norm, 141
n u l l , 139
Vector ( c o n t i n u e d ) ,
o p p o s i t e , 138
orthogonal, 1 5 6
parallel, 143, 163
perpendicular, 1 5 3
representation by located v e c t o r , 14U
r i s e , 137
row, 4 , 136
Vector (continued) ,
r u n , 137
slope,
138
space, 1 6 6
subspace,
166167
terminal p o i n t , 137
variable, 177
Zero matrix, 11