Page 204 - 35Linear Algebra
P. 204
204 Linear Independence
Definition We say that the vectors v 1 , v 2 , . . . , v n are linearly dependent
1
2
n
1
if there exist constants c , c , . . . , c not all zero such that
2
1
n
c v 1 + c v 2 + · · · + c v n = 0.
Otherwise, the vectors v 1 , v 2 , . . . , v n are linearly independent.
Remark The zero vector 0 V can never be on a list of independent vectors because
α0 V = 0 V for any scalar α.
3
Example 115 Consider the following vectors in R :
4 −3 5 −1
v 1 = −1 , v 2 = 7 , v 3 = , v 4 = 1 .
12
3 4 17 0
Are these vectors linearly independent?
No, since 3v 1 + 2v 2 − v 3 + v 4 = 0, the vectors are linearly dependent.
Worked Example
10.1 Showing Linear Dependence
In the above example we were given the linear combination 3v 1 +2v 2 −v 3 +v 4
seemingly by magic. The next example shows how to find such a linear
combination, if it exists.
3
Example 116 Consider the following vectors in R :
0 1 1
2
2
0
v 1 = , v 2 = , v 3 = .
1 1 3
Are they linearly independent?
We need to see whether the system
3
2
1
c v 1 + c v 2 + c v 3 = 0
1
Usually our vector spaces are defined over R, but in general we can have vector spaces
i
defined over different base fields such as C or Z 2 . The coefficients c should come from
whatever our base field is (usually R).
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