Vector Worksheets With Answers - Math 125, Exam 3 Page 11
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3
1
6
1
2. Consider the matrix A =
.
2
6
2
4
2
1
3
1
6
3
(a) (2 pts) Give a definition of the column space of a matrix.
Solution: The column space is the span of the column vectors a
, a
, a
, a
, and a
. (Sim-
1
2
3
4
5
ilarly, all linear combinations of a
, a
, a
, a
, and a
.)
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3
4
5
(b) (5 pts) Use a dependency table to select a basis for the column space of A and write
any extraneous columns as a linear combination of the basis.
Solution: First, we create the initial dependency table using matrix A:
a
a
a
a
a
1
2
3
4
5
e
1
3
1
6
1
1
.
e
2
6
2
4
2
2
e
1
3
1
6
3
3
Moving the matrix into reduced row echelon form, we get:
a
a
a
a
a
1
2
3
4
5
a
1
3
0
2
0
1
.
a
0
0
1
4
0
3
a
0
0
0
0
1
5
We see that the set a
, a
, a
form a basis for column space.
1
3
5
Reading down the columns of vectors not in the basis, we see that a
and a
can be
2
4
written as a
=
3 a
and a
= 2 a
+ 4 a
.
2
1
4
1
3
(c) (2 pts) What is the dimension of the column space? Explain your answer.
Solution: Since there are three vectors in the basis of the column space of A, we know that
the dimension is three.
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