Math 222-500
Solutions Exam 1
October 10, 2006
1
.
(15) Define the following:
a
.
the span of the set of vectors
x
u
1
,
x
u
2
,
u
,
x
u
k
,
The set of all linear combinations of these vectors is the span.
b
.
the set of vectors
x
u
1
,
x
u
2
,
u
,
x
u
k
is linearly independent.
The set is linearly independent if whenever
c
1
x
u
1
U
u
U
c
k
x
u
k
±
0
u
then each of the
c
i
must equal zero.
c
.
the null space of an
m
u
n
matrix
A
.
The null space of
A
is the set of vectors
x
u
u
R
n
such that
Ax
u ±
0
u
.
2
.
(25) A system of equations has
A
as its coefficient matrix, and
A
is row equivalent to the matrix
B
±
1 0
2
1 0 1
0 1
U
1 0 0 2
0 0
0
0 1 1
0 0
0
0 0 0
.
a
.
Find a basis for the null space of
A
.
The matrix
B
tells us that
x
5
±
U
x
6
,
x
2
±
x
3
U
2
x
6
, and
x
1
±
U
2
x
3
U
x
4
U
x
6
.
That is, we have 3 bound variables (
x
1
,
x
2
, and
x
5
) with the other 3 being free variables. Thus,
the null space has dimension 3, and basis equal to
U
U
2,1,1,0,0,0
±
,
U
U
1,0,0,1,0,0
±
,
U
U
1,
U
2,0,0,
U
1,1
±
.
b
.
What is the dimension of the column space of
A
?
The dimension of the column space is the number of free variables, which is 3.
c
.
What is the dimension of the row space of
A
?
The dimension of the row space is the same as the dimension of the column space. Thus, it
too is 3.
d