Homework 5, Due April 1
Adam Gamzon
Related reading: sections 3.4, 4.1, and 4.2 from the text
1. Let
γ
=
1
2
,
5
6
and let
β
=
2
5
,
5
12
. These are bases of
R
2
. (You don’t need to
prove this.)
(a) Let
v
=

4
4
. Compute [
v
]
γ
.
Solution:
Solve the system represented by the matrix:
1
5


4
2
6

4
.
This has rref
1
0

11
0
1


3
,
so [
v
]
γ
=
11

3
.
(b) Determine the change of basis matrix, [1]
βγ
, that takes
γ
coordinates to
β
coordinates.
Solution:
The change of basis matrix is
2
5
5
12

1
1
5
2
6
=

12
5
5

2
1
5
2
6
=

2

30
1
13
.
(c) Use part (b) to find [
v
]
β
.
Solution:
[
v
]
β
=
[1]
β
γ
[
v
]
γ
=

2

30
1
13
11

3
=
68

28
2. Consider the vector space
P
2
=
{
ax
2
+
bx
+
c
:
a, b, c
∈
R
}
. (Remember, the symbol
∈
means
“in.”)
(a) Show that
β
=
{
1 +
x
2
,

3
x,
2
x
+
x
2
}
is a basis of
P
2
.
1
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Solution:
Since dim
P
2
= 3, it is enough to show that these three vectors are linearly
independent. That is, we’ll show that the only real numbers
a, b,
and
c
such that
a
(1 +
x
2
) +
b
(

3
x
) +
c
(2
x
+
x
2
) = 0
are
a
=
b
=
c
= 0. Looking at the equation, we get a system of linear equations
a
=
0
(

3
b
+ 2
c
)
x
=
0
(
a
+
c
)
x
2
=
0
,
or equivalently,
a
=
0

3
b
+ 2
c
=
0
a
+
c
=
0
.
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 Spring '10
 STAFF
 Math, Linear Algebra, Algebra, Vector Space

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