MATH 223, Linear Algebra
Winter, 2010
Assignment 3, due
in class
Monday, February 1, 2010
1. Let
V
=
M
2
(
R
) be the real vector space of 2
×
2 matrices with real entries.
For each of the following subsets of
V
, decide if it is independent, if it is
a spanning set for
V
, and/or if it is a basis for
V
. Justify your answers.
(a)
{
1
3
2

1
¶
,
2
0
5
0
¶
,
4
1
0

1
¶
}
.
(b)
{
1
1
1
1
¶
,
1
1

1

1
¶
,
1

1
1

1
¶
,
1
2
3
5
¶
}
.
(c)
{
1
0
0
0
¶
,
1
1
0
0
¶
,
1
1
1
0
¶
,
1
1
1
1
¶
,
0
1
1
1
¶
}
.
2. Find a basis for each of the null space, row space and column space of the
following matrix over
C
.
A
=
1
2

i

3 + 2
i
3
i
0
0

1 +
i
2

2
i

2

3 +
i
1 +
i
2

3 +
i

3 +
i

1

3
i
2
i
3 + 6
i

6

10
i

7 + 3
i
6 + 5
i
.
Express each row of
A
as a linear combination of the vectors in your basis
for the row space, and express each column of
A
as a linear combination
of the vectors in your basis for the column space.
3. In this problem we suppose that
F
is a field,
A
is an
m
×
n
matrix over
F
and that
W
is a subspace of
F
m
.
(a) Show that
U
=
{
~v
∈
F
n
:
A~v
∈
W
}
is a subspace of
F
n
.
(b) Now suppose that
m
=
n
and
A
is invertible, and that
B
=
{
~v
1
, . . . ,~v
k
}
is a basis for
W
. Show that
{
A

1
~v
1
, . . . , A

1
~v
k
}
is a basis for
U
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 Winter '10
 Loveys
 Linear Algebra, Algebra, Matrices, Vector Space, Sets, real vector space, real number r.

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