M341 Midterm Exam 2
56215
Solution
I. (20 points) Mark the following statements either "True" or "False". Circle T for
true and F for False. No proof or explanation is needed.
1. (T) If
A
,
B
are nonsingular
n
×
n
matrices, then
((
AB
)
T
)
−
1
= (
A
−
1
)
T
(
B
−
1
)
T
.
2. (F) The inverse of a type (I) row operation is a type (II) row operation.
3. (F) Span
(
S
)
is only de ned if
S
is a nite subset of a vector space.
4. (T) An
n
×
n
matrix
A
has determinant zero if and only if rank
(
A
)
< n
.
5. (F) The set of all polynomial of degree 7 is a vector space under the usual operations of
addition and scalar multiplication.
6. (F) Any plane
W
in
R
3
is a subspace of
R
3
under the usual operations.
7. (F) The set of all vectors of the form
[0
, a, b,
1]
is a subspace of
R
4
under the usual
operations.
8. (F) Let
W
1
and
W
2
be two subspaces of a vector space
V
. Then their union
W
1
∪ W
2
is
also a subspace of
V
.
9. (T) If two rows of a square matrix
A
are identical, then

A

= 0
.
10. (T) If
x
is a linear combination of the rows of
A
, and
B
is row equivalent to
A
, then
x
is
in the row space of
B
.
II. Rank
1. (3 points) Give the de nition of the rank of a matrix
A
.
Ans:
The rank of a matrix
A
is the number of nonzero rows in the unique reduced row
echelon form that is row equivalent to
A
.
2. (4 points) Let
A
and
B
be two matrices as following:
A
=
1
−
2
0
0 3
2
−
5
−
3
−
2 6
0
5
15
10 0
2
6
18
8 6
,
B
=
0 2
7
5
0
0 1
3
2
0
−
1 2
1
0
−
3
2 6 18 10
6
Find the rank of
A
and
B
.
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 Spring '08
 HIETMANN
 Linear Algebra, Vector Space, aij, cij

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