(a) Let V = F[x] be the vector space of polynomials over a field F. Show that the setS = {1, x, x², x', ..., } is a basis of V.(b) Let V be the F-vector space of F-valued sequences (a„)n20. For each i > 0, let ô; be the sequencewhose i-th term is 1, and whose other terms are 0. Show that the set 6o, d1, ..., 8n, ... does not spanV, 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 noone 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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(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.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.1: Orthogonality In Rn

Problem 34EQ

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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, x', ..., } is a basis of V.
(b) Let V be the F-vector space of F-valued sequences (a„)n20. For each i > 0, let ô; be the sequence
whose i-th term is 1, and whose other terms are 0. Show that the set 6o, d1, ..., 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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