Page 203 - 35Linear Algebra

10

Linear Independence

3
Consider a plane P that includes the origin in R and non-zero vectors
{u, v, w} in P.

If no two of u, v and w are parallel, then P = span{u, v, w}. But any two
vectors determines a plane, so we should be able to span the plane using
only two of the vectors u, v, w. Then we could choose two of the vectors in
{u, v, w} whose span is P, and express the other as a linear combination of
1
2
those two. Suppose u and v span P. Then there exist constants d , d (not
1
2
both zero) such that w = d u + d v. Since w can be expressed in terms of u
and v we say that it is not independent. More generally, the relationship
i
2
1
3
i
c u + c v + c w = 0 c ∈ R, some c 6= 0
expresses the fact that u, v, w are not all independent.
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