Math 235
Sample Midterm  2
NOTES:  Our Midterm does not cover the GramSchmidt procedure.
1.
Short Answer Problems
a) Write a basis for the four fundamental subspaces of
A
=
1
0
0

1
0
0
1
1
0
0
0
0
.
b) Determine if
h
p, q
i
=
p
(

1)
q
(

1) +
p
(1)
q
(1) is an inner product for
P
2
.
c) State the RankNullity Theorem.
d) Find the rank and nullity of the linear mapping
T
:
P
2
→
M
2
×
2
(
R
) defined by
T
(
a
+
bx
+
cx
2
) =
c
b
0
c
2.
Let
L
:
M
2
×
2
(
R
)
→
M
2
×
2
(
R
) be given by
L
(
A
) =
1
2
3
4
A
T
.
Find the matrix for
L
relative to the basis
B
of
M
2
×
2
(
R
), where
B
=
1
0
0
0
,
0
1
0
0
,
0
0
1
0
,
0
0
0
1
3.
Let
B
=
{
v
1
, v
2
}
be a basis for
V
. Let
a
be a scalar constant. Let
T
:
V
→
V
be linear and
T
(
v
1
) =
av
1
+
av
2
,
T
(
v
2
) = 3
v
1

av
2
. For what values of
a
is
T
an isomorphism?
4.
Let
N
be the plane with basis
1
0

1
,
1

1
1
. Define an explicit isomorphism
to establish that
P
1
and
N
are isomorphic. Prove that your map is an isomorphism.
5.
Let
T
be a linear operator on an inner product space
V
, and suppose that
h
~x, ~
y
i
=
h
T
(
~x
)
, T
(
~
y
)
i
for all
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 Spring '08
 CELMIN
 Math, Linear Algebra, Vector Space, inner product

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