1. For the following subsets of vector spaces, state whether or not the indicated subset is a subspace. Justify your answers by giving a proof or a counter-example in each case. (iv) The set G of all polynomials p(x) with p(1) = p(0), in the vector space P3 of polynomials of degree at most 3 with coecients in R. (v) The set Z of all sequences which are eventually zero, Z = {v = (v0, v1, v2,...) is an element of F^infinity : there is n such that vi = 0 for all i >= n}, in the vector space F^infinity of infinite sequences v = (v0, v1, v2,...) with vi is an element of F (F any field).

1. For the following subsets of vector spaces, state whether or not the indicated subset is a subspace. Justify your answers by giving a proof or a counter-example in each case. (iv) The set G of all polynomials p(x) with p(1) = p(0), in the vector space P3 of polynomials of degree at most 3 with coecients in R. (v) The set Z of all sequences which are eventually zero, Z = {v = (v0, v1, v2,...) is an element of F^infinity : there is n such that vi = 0 for all i >= n}, in the vector space F^infinity of infinite sequences v = (v0, v1, v2,...) with vi is an element of F (F any field).

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.2: Vector Spaces

Problem 37E: Let V be the set of all positive real numbers. Determine whether V is a vector space with the...

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