Math 334 Midterm 1
Do only 3 of the four problems 25.
Do all work and record all answers
in the blue book, with your name on the front.
1. (20 points) a. If
V
is a vector space, define what it mean for a subset
U
⊂
V
to be a
subspace
.
b.
For each of the following subsets of
R
2
, say which of the axioms
of a subspace are satisfied.
i.
V
=
(
x, y
)
x
= 2
⊂
R
2
none of them
ii.
V
=
(
x, y
)
xy
= 0
⊂
R
2
all except closed under addition
2. (30 points) a. Define what it means for a subset
B
⊂
V
to be
linearly
independent
, to
span
V, and to be a
basis
.
b. Let
V
=
P
3
, let
T
:
V
→
V
be the operator
(
Tf
)(
x
) =
f
(
x
+ 1)
,
and consider the basis
B
=
{
1
, x, x
2
} ⊂
V
. Find the matrix
M
B
,
B
(
T
)
of
T
in the basis
B
.
1
1
1
1
2
1
3. (30 points) a. If
U, W
⊂
V
are subspaces, define the sum
U
+
W
.
b.
Suppose
U, W
are subspaces of
R
5
of dimensions 2 and 4.
Prove
that
U
∩
W
contains a vector other than 0.
by a theorem in the book, dim(
U
+
W
) = dim(
U
)+dim(
W
)

dim(
U
∩
W
) = 6

dim(
U
∩
W
). but dim(
U
+
W
)
≤
5 since it’s a subspace of
R
5
, so dim(
U
∩
W
)
6
= 0, meaning it’s not the zero vector space.
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 Spring '11
 Carlsson
 Linear Algebra, Algebra, Vector Space, blue book, null space Null

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