(a) Let V = F[x] be the vector space of polynomials over a field F. Show that the set S = {1, x, x², x³3,...,} is abasis of V.(b) Let V be the F-vector space of F-valued sequences (an)n>0. For each i > 0, let 8; be the sequence whose i-thterm is 1, and whose other terms are 0. Show that the set 8o, 81,.….., 8n , . . does not span V, by giving an explicitdescription of the span of this set.Remark: A general principle of logic known as Zorn's implies that every vector space has a basis. But no one has everbeen (nor will ever be) able to write down an explicit basis for the vector space V of (b).

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(a) Let V = F[x] be the vector space of polynomials over a field F. Show that the set S = {1,x,x², x³,..,} is a basis of V.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.2: Integral Domains And Fields

Problem 25E: Suppose S is a subset of an field F that contains at least two elements and satisfies both of the...

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Question

(a) Let V = F[x] be the vector space of polynomials over a field F. Show that the set S = {1, x, x², x³3,...,} is a
basis of V.
(b) Let V be the F-vector space of F-valued sequences (an)n>0. For each i > 0, let 8; be the sequence whose i-th
term is 1, and whose other terms are 0. Show that the set 8o, 81,.….., 8n , . . does not span V, by giving an explicit
description of the span of this set.
Remark: A general principle of logic known as Zorn's implies that every vector space has a basis. But no one has ever
been (nor will ever be) able to write down an explicit basis for the vector space V of (b).

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