This preview shows pages 1–2. Sign up to view the full content.
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
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '08
 GUREVITCH
 Math, Matrices

Click to edit the document details