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Prove the following infinite-dimensional version of Theorem 1.8 (p. 43): Let ft be a

Chapter 1, Problem 5

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QUESTION:

Prove the following infinite-dimensional version of Theorem 1.8 (p. 43): Let ft be a subset of an infinite-dimensional vector space V. Then ft is a basis for V if and only if for each nonzero vector v in V, there exist unique vectors u\, u2,..., un in ft and unique nonzero scalars C\, c 2 ,... , cn such that v ciui + c2u2 -\ 1- cnu

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QUESTION:

Prove the following infinite-dimensional version of Theorem 1.8 (p. 43): Let ft be a subset of an infinite-dimensional vector space V. Then ft is a basis for V if and only if for each nonzero vector v in V, there exist unique vectors u\, u2,..., un in ft and unique nonzero scalars C\, c 2 ,... , cn such that v ciui + c2u2 -\ 1- cnu

ANSWER:

Step 1 of 4

Suppose is the basis for and . Thus, it can be concluded that,

from

Hence, for vector  linear combination is .

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