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1
Mathematics 21b  Fall 2000  First Exam
This exam was 90 minutes long. Calculators were not allowed.
1)
(12 points) True or False. (Circle one)
You need not give your reasoning.
a)
The kernel of rref(
A
) is the same as the kernel of
A
.
a)
TRUE
FALSE
b)
The image of rref(
A
) is the same as the image of
A
.
b)
TRUE
FALSE
c)
Let
A
and
B
be n x n matrices, with
AB
=
BA
. Then
A
3
B
=
BA
3
.
c)
TRUE
FALSE
d)
There is a 4 x 4 matrix
A
such that image(
A
) and kernel(
A
) are the
same subspace of
R
4
.
d)
TRUE
FALSE
e)
If
A
and
B
are n x n matrices such that the kernel of
A
is contained in
the image of
B
, then the matrix
AB
cannot be invertible.
e)
TRUE
FALSE
f)
If
A
is an invertible 2 x 2 matrix, then the rank of
A
is necessarily 1.
f)
TRUE
FALSE
2)
(18 points) Let
A
be the 4 x 5 matrix
A =
1
2
1
5
0
1
2
1
1
0
3
6
1
7
1
1
2
0
3
0








.
a) Represent the kernel of
A
as the span of a set of vectors.
b) Represent the image of
A
as the span of a set of vectors. Eliminate any vectors which can be
expressed as linear combinations of other vectors in your set. [This is called a basis for im(
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