HOMEWORK 2
1.
(a) Let
S
n
×
n
(
C
) denote the set of all
n
×
n
symmetric matrices with complex entries.
Prove that this set is a subspace of
M
n
×
n
(
C
). Find a basis for this subspace, and
give a formula for its dimension (in terms of
n
).
(b) Recall that for a square matrix
A
, tr(
A
) denotes the sum of the diagonal entries
of
A
. Prove that
{
A
∈
M
2
×
2
(
C
) : tr(
A
) = 0
}
is a subspace of
M
2
×
2
(
C
) Find a
basis for this subspace and compute its dimension.
(c) Let
W
n
=
{
p
(
x
)
∈
P
n
(
C
) :
p
(

1) =
p
(1)
}
. Show that this is a subspace of
P
n
(
C
).
Find a basis for it, and give a formula for its dimension (in terms of
n
).
2.
(a) Let
S
be a collection of sets (that is, every
element
of
S
is itself a set). Then we
can define the
intersection
of the sets in
S
(even if
S
is an infinite collection of
sets), as follows:
\
X
∈S
X
:=
{
x
:
∀
X
∈ S
(
x
∈
X
)
}
Show that if
V
is a vector space and
W
is a subspace of
V
for each
W
in some
collection
S
, then
T
W
∈S
W
is also a subspace of
V
.
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 Fall '08
 GUREVITCH
 Math, Linear Algebra, Matrices, Vector Space

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