MAT
320
Linear Algebra
Final Exam, Section A
Instructions
: This is an openbook, opennote exam. You must work individually. Show all work relevant to
the solution of each problem. i.e. no credit will be given for “just the answers.”
5
Problem
1
.
Let
A
be an
n
×
n
matrix, and
W
the set of eigenvectors of
A
with eigenvalue
λ
. Prove
W
is a subspace of
R
n
.
5
Problem
2
.
A
projection
is a linear transformation satisfying the condition
T
(
T
(
v
)) =
T
(
v
)
. Let
T
be a
projection, and
A
its standard matrix. Prove
A
is singular  and hence,
T
is not invertible.
10
Problem
3
.
Let
W
be a subspace of a vector space
V
, and
W
⊥
=
{
w
∈
W

h
w,v
i
=
0
∀
v
∈
W
}
.
a.
Prove
W
⊥
is a subspace of
V
.
b.
For
V
=
R
3
, with dot product as the deﬁnition of
h·
,
·i
, and
W
=
{
(
1,0,1
)
,
(
0,0,1
)
}
, ﬁnd a basis for
W
⊥
. What is its dimension?
5
Problem
4
.
Let
M
be the vector space of
2
×
2
matrices, and
W
=
{
A
∈
M

trace
(
A
) =
0
}
. Prove
B
,
as deﬁned below, is a basis for
W
.
B
=
±
±
1
0
0

1
²
,
±
0 1
0 0
²
,
±
0 0
1 0
²
²
5
Problem
5
.
Deﬁne
T
:
W
→
W
such that
T
(
X
) =
XA

AX
, where
A
is a ﬁxed
2
×
2
matrix and
W
is the subspace of
M
2
×
2
with the basis
B
as deﬁned in Problem
4
. Prove
T
is a linear transformation,
where
A
is any
2
×
2
matrix.
10
Problem
6
.
With
T
as deﬁned in Problem
5
, and
W
and
B
as deﬁned in Problem
4
:
b.
Let
A
=
±
1
0
0

1
²
. Find the standard matrix for
T
with respect to the basis
B
.
c.
Let
A
=
±
1
0
0

1
²
. Find a basis for the kernel of
T
.
5
Problem
7
.
Consider
R
, the set of all real numbers, and deﬁne
+
so that
e
x
+
e
y
=
e
x
+
y
. Prove
R
,
with this deﬁnition of addition, is a vector space.
5
Problem
8
.
Deﬁne
h
A,B
i
as follows, where
A
and
B
are
2
×
2
matrices.
³±
a
1
a
2
a
3
a
4
²
,
±
b
1
b
2
b
3
b
4
²´
=
a
1
b
1
+
a
2
b
2
+
a
3
b
3
+
a
4
b
4
Prove
h
A,B
i
is an inner product.
1