MATNYC
LAB #5
Linear Dependence, Basis and Dimension
1. Are the following vectors linearly independent? If not, ﬁnd a dependency relationship.
a) [2
,
0
,

1]
,
[3
,

1
,

4]
,
[

1
,
2
,
1]
b) [1
,
1
,
0]
,
[0
,
1
,

1]
,
[

2
,
1
,

3]
,
[1
,
2
,

1]
2. Let
~v
1
,
~v
2
, . . . ,
~v
r
be
r
vectors in
R
n
. Explain how the concept of rank can be used to
determine if the vectors are linearly independent. Give any conditions that must be
satisﬁed by
r
, the number of vectors, in order for the question to be answered positively,
then apply your analysis to determine whether the vectors [1
,
2
,
1
,

3], [

1
,
1
,
2
,
2],
[1
,
5
,
4
,

4] and [1
,
8
,
7
,

4] are linearly independent.
3. Let
~v
1
= [1
,
1
,
1
,
2]
,~v
2
= [1
,

1
,
2
,
3]
,~v
3
= [0
,

2
,
1
,
1]
,~v
4
= [1
,

2
,
3
,
2] and
~v
5
= [0
,

1
,
1
,

1]. Find a subset of
{
~v
1
,~v
2
,~v
3
,~v
4
,~v
5
}
which is a basis for their
linear
span
W
. Write the deleted vector(s) as linear combinations of the basis vectors. What
is the dimension of