Page 397 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

(12)

which, by the linear independence of , implies that

Proof of Theorem 5.4.4b Assume that is a set of vectors in V, and to be specific, suppose that is a linear

combination of , say

(13)

We want to show that if is removed from S, then the remaining set of vectors still

spans span(S); that is, we must show that every vector w in span(S) is expressible as a linear

combination of . But if w is in span(S), then w is expressible in the form

or, on substituting 13,

which expresses as a linear combination of .

Exercise Set 5.4

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Explain why the following sets of vectors are not bases for the indicated vector spaces. (Solve this problem by inspection.)
1.

(a)

(b)

(c)

(d)

Which of the following sets of vectors are bases for
2.

(a) (2, 1), (3, 0)

(b) (4, 1), (−7, −8)