Math 304 Solutions 5
1
Section 3.6
6. How many solutions will the linear system
A
x
=
b
have if
b
is in the
column space of
A
and the column vectors of
A
are linearly dependent.
Explain.
Answer:
There will be infinitely many solutions.
Proof.
By Theorem 3.6.2 in the book (or by the theorem in class), if
b
is in the column space of
A
then the system
A
x
=
b
is consistent.
Thus, the system has either 1 solution or infinitely many solutions.
Since the column vectors are linearly dependent, we know that if we
row reduce the matrix
A
, there will be free variables. Thus, the system
has either no solutions or infinitely many solutions.
Therefore, the system has infinitely many solutions.
9. Let
A
and
B
be row equivalent matrices.
(a) Show that the dimension of the column space of
A
equals the
dimension of the column space of
B
.
(b) Are the column spaces of the two matrices necessarily the same?
Justify your answer.
Answer:
(a) Since
A
and
B
are row equivalent, they have the same row space. Thus,
the dimension of the row space of
A
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 Spring '08
 STAFF
 Linear Algebra, column space

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