Math 2270 Assignment 5
-
Dylan Zwick
Fall 2012
Section 3.1 1, 2, 10, 20, 23
Section 3.2 1, 2, 4, 18, 36
-
-
3.1 Spaces of Vectors
-
In the definition of a vector space, vector addition x + y and scalar multi
plication cx must obey the following eight rules:
(1) x+y=y+x
(2) x+(y+z)=(x+y)+z
(3) There is a unique “zero vector” such that x + 0
=
x for all x
(4) For each x there is a unique vector —x such that x + (—x)
(5) 1 times x equals x
)x
2
(6) (cic
=
(c
1
c
x
2
)
(7) c(x+y)=cx+cy
)x
2
(8) (ci + c
=
x
1
c
x.
2
+c
1
=
0
2 + yr). With the
3.1.1 Suppose (xi, x
) + (yi, y2) is defined to be (x + Y2, x
2
which
of
the
eight
conditions are
multiplication
cx
=
(cu,
)
2
cx
,
usual
not satisfied?
7
(IYLt
(y(fy)
4
=
±((
L)
(
+/
YLYf)
x*y
‘3
—
54-(1
1IG
-
y
1
((
,L)+
(i/L)
+)
2
y
((L
Yt,)
1)
2
j7o
t?)
(3f/e
3.1.2 Suppose the multiplication cx is defined to produce (cxi, 0) instead
2 are the eight conditions
). With the usual addition in 1R
2
of (cxi, cx
satisfied?
±
50/
tk
(, )
9
C6ñ6(/O
rY
5
3
(IX//
)
(, o)
3.1.10 Which of the following subsets of R are actually subspaces?
1
,b
2
,b
1
) with b
3
(a) The plane of vectors (b
1
(b) The plane of vectors with b
3
2
1
(c) The vectors with b
=
=
1.
=
(1.4. 0) and w
1+b
3
2+b
(e) All vectors that satisfy b
4)
c)
<
2
b
<
.
3
b
tJ2cp’(ce
AJo C(
/
ace
d)
4-
b
.
2
b
0.
(d) All linear combinations of v
1
(f) All vectors with b
=
a
4
=
0.
=
(2. 2, 2).
, b2, b
1
) are these systems
3
3.1.20 For which right sides (find a condition on b
solvable?
/1
(a)
f
2
2(x\
8
4
)(
—4 —2)
/
(b)
4
1
1
2
—1
4
\
9
4)
/
(
)
x
/b
1
b
—
f
b3
3)
/b
:r
=
2J
I
3
b
I
hL
o o
)25
O
SC)
/
/
C(
(
H b\
/
5
° 4L17,
0
I Li b
01
00 b
3
h
1
)
/
3.1.23
/
d an extra column b to a matrix A, then the column space
Give an example where the column
gets larger unless
space gets larger and an example where it doesn’t. Why is Ax = b
solvable exactly when the column space doesn’t ger larger it is the
same for A and (A b]?
-
h
ke
5
1
1
Cd1
5
5
1
am
ihe
()
m,9Ie
Co/1n
(qry:
6O5
A
7(0
osii
fhd
c
1
€Ci
o
i
.
,
ai
d
6
0
3.2 The Nulispace of A: Solving Ax
-
=
b
3.2.1 Reduce the matrices to their ordinary echelon forms U:
(1 2246
1 2 3 6 9
(a) A=
\0 0 1 2 3
(
/2 4 2
(b)B=( 044
\o 8 8
c)
II
t23cf)
I L?
() Dl
7
tL3J
3
ra
b)
(
(7
f
1
a3)
3I
0000O
fcc
1wL
Z/ /
zq
oLiy
7
3.2.2 For the matrices in Problem 3.2.1, find a special solution for each free
variable. (Set the free variable equal to 1. Set the other free variables
equal to zero.)
c)
C
fjV
Oo
Cc(fl5
f cfI /
[?5
Cc) [ci1Il J
oo0c)J
ai
(o/c47%J
Ire e
yz_J
j;;E;Z—
1)
)cj)
X I
0)
U
0
-
1Q/ K
—Q
5
*
5
y
-3
.Y
3
b)
(
XLO
CIL
yI/
PV
4L1
C-)
/
XLl,
X(
8
=6;)
L
2/ ,5:
3.2.4 By further row operations on each U in Problem 3.2.1, find the re
duced echelon form R. True or false: The nullspace of R equals the
nullspace of U.
(T
Oo13)
Oc) c)od
b)
4
I
O
0
L
0-/
LHf
ii
Oo
0
MtA)
9
/d)
6).
z = 0
z = 12 is parallel to the plane x
3.2.18 The plane x
in Problem 3.2.17. One particular point on this plane is (12, 0. 0). All
points on the plan have the form (fill in the first components)
—
—
—
—
(x
()
(3
1=10 JI 1
\o)
\oJ
zJ
I
‘
10
3.2.36 How is the nulispace N(G) related to the spaces N(A) and N(B), if
/ ) (Z
(k)
d
,
2c)
/)
4,)
mA)
11
=
j
o /y
1
1
Ac-
cM
q
4ernq1i