1
Lecture 3
3.1 Linear dependence
Let X = {
1
v
,
2
v
, …,
k
v
} be a set of vectors in a vector space V. If
r
1
1
v
+
r
2
2
v
+ … +
r
k
k
v
=
0
for some
0
j
r
,
j
= 1, 2, …, k,
then X is a linearly dependent
set. Otherwise, X is a linearly independent
set.
Example:
Is the set {sin
2
x, cos
2
x,1} linearly dependent or independent?
Examples: In each case determine if the set of vectors is independence.
{
0
1
2
1
,
3
1
0
2
,
6
5
6
1
}, {e
x
, e
2x
}, {1, x, x
2
, …, x
n
, …}
Example:
Let V denote a vector space. Prove:
1.
If
0
v
in V, then {
v
} is an independent set.
2.
No independent set of vectors in V can contain the zero vector.
3.
If {
1
v
,
2
v
, …,
k
v
} is independent in V, so is {
a
1
1
v
,
a
2
2
v
, …,
a
k
k
v
} if
0
i
a
.