M340L Third Midterm Exam, April 7, 2003
1. The following two 4
×
4 matrices are rowequivalent:
A
=
1
1
2
1
1
3
8
0
2
4
10
1
3
5
12
0
;
B
=
1
0

1
0
0
1
3
0
0
0
0
1
0
0
0
0
a) What is the rank of
A
?
b) Find a basis for the column space of
A
.
c) Find a basis for the null space of
A
.
d) Find a basis for the row space of
A
.
2. Let
V
=Span
{
1
1
1
1
,
1
2
3
4
,
1
3
5
8
,
4
3
2
1
} ⊂
R
4
.
a) What is the dimension of
V
?
b) Find a basis for
V
.
c) Let
W
be the subspace of
P
3
[
t
] spanned by the polynomials 1+
t
+
t
2
+
t
3
,
1 + 2
t
+ 3
t
2
+ 4
t
3
, 1 + 3
t
+ 5
t
2
+ 8
t
3
and 4 + 3
t
+ 2
t
2
+
t
3
.
What is the
dimension of
W
? Find a basis for
W
.
3. Consider the following basis for
R
3
:
B
=
1
2
2
,
3
7
7
,
6
12
13
a) Find the changeofbasis matrix
P
EB
that converts from coordinates in
the
B
basis to coordinates in the standard (
E
) basis.
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 Spring '08
 PAVLOVIC
 Linear Algebra, Vector Space, basis, changeofbasis matrix

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