Math 2270 Assignment 1
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Dylan Zwick
Fall 2012
Section 1.1 1,2,13,16,30
Section 1.2 1,2,3,27,29
Section 1.3 1,2,6,8,13
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-
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1
Section 1.1 Vectors and Linear Combinations
1.1.1
escribe geometrically (line, plane, or all of 1R
) all linear combina
3
tions of
-
/3
(a) (\2)and(6
1
JUT5
UT M
—
A
M
+
A
()
12!)
/q2
f17L
1’f
E=9
uio
I
aujd fix
(T)
=
A
M1U Z11
=
4J(
2
7LI
/h2J
iVD]J
20
)
pui
‘
(
1uiJc
)
o
pu
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(1
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1)
7
(
ci
tO’
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(q)
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1.1.13 (a) What is the sum V of the twelve vectors that go from the center
of a clock to the hours 1:00, 2:00,
,4(l
...,
12:00?
( 4—c
Vcc+o
r
(b) If the 2:00 vector is removed, why do the 11 remaining vectors
add to 8:00?
o4e
T:G
ci/
eCo
Ic
of7a5!
ic
i
/
(c) What are the components of that 2:00 vector v (cos
c
2:00 ,
1OQ
rcM );C (X-cI)r)
,
sin 6)?
+
fro 1a;ác(y-cr,J.
V
Sc!
e=
6
(c05y
1.1.16 Mark the point —v + 2w and any other combination cv + dw wil
1. Draw the line of all combinations that have c + ci = 1.
c+d
-U
,‘i
-
7
Li
I
3
cc i1
o4-
flcit 1C’Yf
1.1.30 The linear combinations of v = (a, b) and w = (c, d) fill the plane
bc
Find four vectors u, v, w, z with four com
unless c2
ponents each so that their combinations cu + dv + ew + fz produce
all vectors (b
.b
1
.b
2
.b
3
) in four-dimensional space.
4
.
2
Section 1.2 Lengths and Dot Products
-
1.2.1 Calculate the dot products u v and u w and u (v + w) and w v:
13N
.8
-
iqo
4
/8
cii
ji
—
II
—,
0
-c
cç
CD
0
0
0
n
(D
(D
-I
C
(D
Tj
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oZ
cc
CD
CD
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o •
DoD
cii
N
A
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A
ON
—C
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1
=
n
CD
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IA <
CD
—n
—CD
.CiD
—-
—
IA
1.2.27 (Recommended) If lvii = .5 and wi = 3, what are the smallest and
largest values of lv wil? What are the smallest and largest values
ofv•w?
—
Laryej
i
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jrcc
—
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7
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jffs
Y’77”//-ecj-
t
Lrye
15,
1.2.29 Pick any numbers that add to x + g + z = 0. Find the angle between
your vector v
(x, y, z) and the vector w = (z, x, y). Challenge
v w/ilvllllwii is always
why
question: Explain
—.
;
C
ii
I
-x--Y
/,/}
-x-y ,y)
x-/+Ky
-xt7
1 =
/yly
t
xy
veco1
Li
o /uJtyr
b
II1
L
6
h /r o
(\/, j x),
i14
cMf
0
1
i-
ii
I
—
.
(ID
II)
cj
(N
+ U
(ID
CC
C
(ID
U
C
—
C
II
(N
II
(I)
(\
these equations Sy
/i
1
1
0
1
1
(
Y2
,s
1
,s
2
3 in the columns of S:
b with s
/i\
(Yi)
0
1
o)
=
=
/i 00) (Yi\
(
1
and
\1J
1
1
1
1
0
1
Y2)
=
The sum of the first n odd numbers is
/
I
i/C
I
(y4\/
(
\7):
) t)
yl
yI
YYz
8
/i\
4
(\9J
1.3.6 Which values of c give dependent columns (combination equals zero)?
(
3 5\
/i o \ /c c
124),(11O),(215).
1 cJ \O 1 1/ \3 3 6J
i
t+
Lhy
br
C7
(3
rtri
1
ic
e
c.
O
5ecJ
I
)
1/
1
33
/
9
h
/
e Ate
(C
(5)
31J
1.3.8 Moving to a 4 by 4 difference equation Ax = b, find the four com
Sb to find the
,
1
. 14. Then write this solution as x
3
x
ponents x
inverse matrix S A’:
Ax=
1
—1
0
0
000
100
—1 1 0
0 —1 1
12
13
14
,23
I
/(
11
=b.
)
1
x
xt1
—
b
1
2
b
3
b
4
b
=
c
ii
O()
1
10
L/
1.3.13 The very last words of worked example 1 .3B say that the 5 by 5 cen
tered difference matrix is not invertible. Write down the 5 equations
Cx = b. Find the combination of left sides that gives zero. What
5 must be zero? (The 5 columns lie on a
,b
1
,b
2
,b
3
,b
4
combination of b
in 5-dimensional space.)
hyperplane”
“4-dimensional
(oc
C
0
(U —1 0I
C) —10 /
() C) -1
01
-
-y;
y
Ct
/ jñ
t4
)
So
o4
F
11
5
0’