Math 222  200
Exam 1 Solutions
February 25, 2005
1. (20) Define the following terms:
(a) linearly independent set of vectors
A set of vectors
{
~x
1
,
· · ·
,
~x
k
}
is linearly independent if whenever
λ
1
~x
1
+
· · ·
+
λ
k
~x
k
=
~
0,
then
λ
1
=
λ
2
=
· · ·
=
λ
k
= 0.
(b) subspace
A nonempty subset
S
of a vector space
V
over a field
F
is called a subspace if
i.
~a
and
~
b
in
S
implies
~a
+
~
b
is also in
S
ii.
α
∈
F
and
~a
∈
S
implies that
α~a
∈
S
.
(c) dimension of a vector space
The dimension of a vector space
V
is the number of vectors in a basis of
V
.
(d) null space of a matrix
If
A
is an
m
×
n
matrix, the null space of
A
is the set of vectors
~x
∈
R
n
(thought
of as
n
×
1 matrices) such that
A~x
=
~
0
∈
R
m
.
2. (30) For the system of equations below:
2
x
1

x
2
+ 3
x
4
=
1
x
2

x
4
=
6
Before answering any of the following questions the reduced row echelon form of the
augmented matrix of this system is computed
ˆ
A
=
2

1
0
3
1
0
1
0

1
6
=
⇒
1
0
0
1
7
/
2
0
1
0

1
6
(a) Find a particular solution.
The given system is equivalent to the system
x
1
+
x
4
=
7
/
2
x
2

x
4
=
6 .
Setting
x
3
=
x
4
= 0, we have the particular solution
h
7
/
2, 6, 0, 0
i
.
(b) Find a basis for the null space of the coefficient matrix.
The coefficient matrix of the system is row equivalent to
1
0
0
1
0
1
0

1
.
There are 2 free variables
x
3
and
x
4
, which tells us that the null space has dimension
two. A basis for the null space is
{h
0, 0, 1, 0
i
,
h
1, 1, 0, 1
i}
.
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(c) Find bases for the row and column spaces of the coefficient matrix.
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 Spring '08
 Stecher
 Linear Algebra, Algebra, Vectors, Vector Space, Linear combination

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