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MATH 223, Linear Algebra
Winter, 2010
Assignment 5, due
in class
Monday, February 15, 2010
Reminder: the midterm exam is Thursday, March 4, from 67PM. The rooms
in the Stewart Biology building are N2/2 and S1/3.
1. Let
V
=
P
3
(
x
), the space of real polynomials with degree at most 3. Deﬁne
T
:
V
→
V
by
T
(
p
(
x
)) = (
x
2

2
x
)
p
00
(
x
) + (2
x
+ 2)
p
0
(
x
)

12
p
(
x
)
.
(a) Verify that
T
is a linear operator on
V
.
(b) Find [
T
]
B
where
B
= (1
,x,x
2
,x
3
) is the standard ordered basis for
V
.
(c) Find [
T
]
B
0
where
B
0
= (1
,x
+1
,x
3

x
2
,x
2
) is a nonstandard ordered
basis for
V
.
(d) Find a basis for each of
ker
(
T
) and
im
(
T
).
(e) Show that
T
(
T
+ 12
I
)(
T
+ 10
I
)(
T
+ 6
I
) = 0.
2. Let
V
be the vector space
M
n
(
F
) of all
n
×
n
matrices with entries from
the ﬁeld
F
; let
P
∈
V
be any particular invertible matrix. Show that if
we deﬁne
TX
=
X

P

1
XP
for every
X
∈
V
, then
T
is a linear operator
on
V
.
For the rest of the problem, assume that
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 Winter '10
 Loveys
 Linear Algebra, Algebra

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