Midterm Solutions
Josh Hernandez
November 4, 2009
1
(20 points) Compute the matrix representation [
T
]
γ
β
of the linear transformation
T
:
R
2
→
R
2
with mapping
T
(
x, y
) = (
x

y,
2
x
+
y
)
with respect to the ordered bases
β
=
{
(1
,
0)
,
(0
,
1)
}
and
γ
=
{
(1
,
1)
,
(1
,
0)
}
.
Solution:
First compute
T
(
β
):
T
(1
,
0) = (1

0
,
2(1) + 0) = (1
,
2)
and
T
(0
,
1) = (0

1
,
2(0) + 1) = (1
,
1)
.
Now, we write each of those vectors as linear combinations in
γ
:
(1
,
2) = 2(1
,
1) + 3(1
,
0)
and
(1
,
1) = 1(1
,
1) + 0(1
,
0)
.
We write these coefficients vertically as the columns of our matrix:
[
T
]
γ
β
=
2
1
3
0
.
2
(20 points) Define
W
=
{
f
∈
P
3
(
R
)

f
0
(0) = 0
}
a
Prove that
W
is a subspace of
P
3
(
R
).
Solution:
Take
f, g
∈
W
and
k
∈
R
. By linearity of derivatives,
(
f
+
kg
)
0
(0) =
f
0
(0) +
kg
0
(0) = 0 +
k
0 = 0
.
Thus
f
+
kg
∈
W
, and we have proved the subspace criterion.
b
Find a basis for
W
(and prove that it is a basis).
Solution:
To find the generic element of
W
, we start with the generic element of
P
3
(
R
) and apply
the constraints. Define
f
(
x
) =
a
+
bx
+
cx
2
+
dx
3
. If
f
∈
W
, then
0 =
f
0
(0) = (
b
+ 2
cx
+ 3
dx
2
)

x
=0
=
b
Thus
W
=
{
a
+
cx
2
+
dx
3
:
a, c, d
∈
R
}
.
Let
β
=
{
1
, x
2
, x
3
}
. Clearly this is a spanning set – given
f
∈
W
,
f
=
a
+
cx
2
+
dx
3
=
a
(1) +
c
(
x
2
) +
d
(
x
3
)
.
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 Spring '11
 Hernandez/Rag
 Linear Algebra, Vector Space, basis, Linear combination, Φβ

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