Math 2270 Assignment 13
-
Dylan Zwick
Fall 2012
Section 7.1 1, 3, 4, 10, 16
Section 7.2 5, 14, 15, 17, 26
Section 7.3 1, 5, 6, 7, 9
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-
-
1
7.1 The Idea of a Linear Transformation
-
7.1.1
A linear transformation must leave the zero vector fixed: T(O)
0. Prove this from T(v + w) = T(v) + T(w) by choosing w
Prove it also from T(cv) = cT(v) by choosing c
-
0.
r( T(j
—___)
T
/
—
1LQ)()
T()= T(o)
2
or(
=
=
=
Nhich of these transformations are not linear? The input is v
.
1
-v
(a) T(v)
.vi)
2
(v
(b) T(v)
(vi.vi)
(c) T(v)
(O,vi)
(d) T(v)
(0, 1)
(e) T(v)
=
1
V
(f) T(v)
=
2
1
v
—
2
V
L
ieY
Ct
ct /
AJ 1,ectY
e
L
/V&+ I(,lecV
3
=
7.1.4
-
IfS and Tare linear transformations, is S(T(v)) linear or quadratic?
(a) (Special case) If S(v)
=
v and T(v)
=
v, then S(T(v))
=
v or v
?
2
) = S(w
2
) + S(w
1
) and T(vi + v
2
)
2
(b) (General case) S(wi + w
T(vi) + T(v
) combine into
2
S(T(vi + V2)) = S(
(1( ,))
) +T() )
1
T(V
=
=
+________
$(T())
5
(i (
5 (i (
,
4
j)
5 ( ( z ii
-
7.1.10
A linear transformation from V to W has an inverse from W to V
when the range is all of W and the kernel contains only v = 0. Then
T(v) = w has one solutions v for each w in W. Why are these T’s not
invertible?
-
2
W=R
v
)=(v
T(v,v
,
2
(a) )
)
2
(b) T(vi, v
=
W
(vi, v
,v
2
1 +v
)
2
3
1R
W=IR’
)=vi
2
v
1
(c) T(v
I
OLkJ
ii
44
/c
I
/k
7
all
H, I)?
=
I)
o
r
i•)
F
I)
T(o,
5
qj1f
7.1.16
Suppose T transposes every matrix Al. Try to find a matrix A
which gives AM = MT for every Al. Show that no matrix A will do
it. To professors: Is this a linear transformation that doesn’t come from
a matrix?
-
4
4*
t
(
-
0
oJ
-
q,,
T
e
Lf
rn
/ I
-
rna/r;
/i
/
/
q/ly
1
( 0
—
OOj
10C)
0
o
6
q
7.2 The Matrix of a Linear Transformation
-
7.2.5
) w
1
2 and T(v
) =
2
,w
2
, suppose T(v
3
,v
1
,v
2
3 and w
,w
1
With bases v
) =w
3
T(v
1 +w
. T is a linear transformation. Find the matrix A and
3
multiply by the vector (1, 1, 1). What is the output from T when the
?
3
1+v
2+v
input is v
-
()
I(
z
k
7
,
(a) What matrix transforms (1.0) into (2.5) and (0.1) to (1.3)?
(b) What matrix transforms (2, 5) to (1, 0) and (1. 3) to (0. 1)?
(c) Why does no matrix transform (2, 6) to (1, 0) and (1, 3) to (0, 1)?
/ H
7 ‘r’
c)
L5i
(,6) Z(1
i)
1
A
Io)
7/ó
8
4
(fe)
7.2.15
-
(a) What matrix M transforms (1, 0) and (0, 1) to (r, t) and (s, u)?
(b) What matrix N transforms (a, c) and (b, d) to (1, 0) and (0, 1)?
(c) What condition on a, b, c, d will make part (b) impossible?
H-b
-c
()
9
7.2.17
If you keep the same basis vectors but put them in a different
)- ‘i
order, the change of basis matrix M is a
rrn &t
their
lengths,
matrix. If you keep the basis vectors in ordei’but change
matrix.
M is a
,
/-
I
10
Suppose v
,v
1
,v
2
3 are eigenvectors for T. This means T(v) =
for i = 1, 2, 3. What is the matrix for T when the input and output
bases are the v’s?
7.2.26
-
/
023
11
7.3 Diagonalization and the Pseudoinverse
-
7.3.1
-
(a) Compute ATA and its eigenvalues and unit eigenvectors vi and
. Find
2
v
Rank one matrix
A
=
11 2
6
(b) Compute AAT and its eigenvalues and unit eigenvectors ui and
.
2
U
1
(c) Verify that Av
=
. Put numbers into the SVD:
1
u
A=UZVT
/1 2\
3 6)
=
(
1
u
2
U
)
(
(
o1
cJ
(
\2
YO/Z
Zc
/0
(0
o-
(
lo
/
o qo)(
i5c
)
): o)
12
2
V
)
T
/10
(z Ye
(0-A
10
vi
L
I’
fr
I
j
H
4
i)
ci
Ii’
I
—
j
—
I
—
CD
CD
a
CD
a
CD
C”
CD
CD
a
C”
CD
CD
I
I.
cE
—
o
ct
-d
•
CiDd
Zc
“—4
ct
“--I
co
4
-J-
1
(__
—
-
-_
—
S
—
-k
rv-
-
—/
)
-
r
--J
¶• ¶
HH
—---------
Is
1>c\
>
cJ
I’
>J
7.3.6
-
AAT has the same eigenvalues u and cr as ATA. Find unit eigen
vectors u
1 and u
. Put numbers into the SVD:
2
33
i)=(ui u
)(
2
1
A=(
AA
(33)
(
U
)
2
U
(vi v
)
2
Io
-I)
( )
0
/(
/o
1(Co
/to)(m
O)(
01
15
)
T
v+
1
7.3.7 In Problem 6, multiply columns times rows to show that A = u
=
UVT
that every matrix of rank r is the sum
v. Prove from A
2
u
one.
matrices
of
rank
of r
T
({) ( )
(oo)
?4 k)
I
6
16
;
°
7.3.9 The pseudoinverse of this A is the same as
A
17
A
because