HOMEWORK 4
1. Compute the determinant of

2 +
i

1
5
i
3
3 + 2
i

2
i
4
i
0
1 +
i
.
2. We say that matrices
A, B
∈
M
n
×
n
(
k
) are
similar
iff there exists an invertible
C
∈
M
n
×
n
(
k
) such that
A
=
CBC

1
. Show that if
A
and
B
are similar matrices, then they
have the same:
(a) trace [Hint: You may use the results of problem 7 below without proof];
(b) determinant;
(c) nullity; and
(d) rank.
where the nullity of a matrix
A
is the nullity of the corresponding transformation
L
A
,
and likewise for rank.
3. Let
T, U
:
V
→
W
be isomorphisms, and
α
∈
k
\ {
0
}
a nonzero scalar.
(a) Is
T
+
U
:
V
→
W
also an isomorphism? Prove or give a counterexample.
(b) Is
αT
:
V
→
W
also an isomorphism? Prove or give a counterexample.
4.
(a) Suppose that
T
:
V
→
W
and
U
:
W
→
X
are invertible linear transformations.
Show that
UT
:
V
→
X
is also invertible, and that (
UT
)

1
=
T

1
U

1
.
(b) Suppose that
A, B
∈
M
n
×
n
(
k
) are invertible matrices. Show that
AB
is invertible,
and that (
AB
)

1
=
B

1
A

1
.
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 Fall '08
 GUREVITCH
 Math, Linear Algebra, Determinant, Matrices, Vector Space, dimensional vector spaces

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