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. 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. of the vector space R³. a b -{{"+") CRYIMAER} a, b 2a 3b/ (i) The subset U = (ii) The subset V = {(0) € R³: a+b+c=1 of the vector space R³. (iii) The set D of matrices of determinant 0, in the vector space M2×2 (R) of all real 2×2 matrices. (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 coefficients in R. (v) The set Z of all sequences which are eventually zero, Z = {v = (vo, V1, V2, ... ..) € F∞ : there is n such that vi = 0 for all i ≥n}, in the vector space F of infinite sequences v = (vo, V₁, V2,...) with v; E F (F any field).

. 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. of the vector space R³. a b -{{"+") CRYIMAER} a, b 2a 3b/ (i) The subset U = (ii) The subset V = {(0) € R³: a+b+c=1 of the vector space R³. (iii) The set D of matrices of determinant 0, in the vector space M2×2 (R) of all real 2×2 matrices. (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 coefficients in R. (v) The set Z of all sequences which are eventually zero, Z = {v = (vo, V1, V2, ... ..) € F∞ : there is n such that vi = 0 for all i ≥n}, in the vector space F of infinite sequences v = (vo, V₁, V2,...) with v; E F (F any field).

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.1: Vector Spaces And Subspaces
Problem 49EQ
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. 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. ER³: a, b ER of the vector space R³. c=1} (i) The subset U (ii) The subset V = = a+b {(2+2) b 2a + 3b a {0) C € R³: a+b+c=1 of the vector space R³. (iii) The set D of matrices of determinant 0, in the vector space M2×2 (R) of all real 2×2 matrices. (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 coefficients in R. (v) The set Z of all sequences which are eventually zero, Z = {v = (vo, V₁, V2, ...) € F∞: there is n such that vi in the vector space F of infinite sequences v = (vo, V₁, V2, ...) with v; ¤ F (F any field). 0 for all i≥n},
Transcribed Image Text:. 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. ER³: a, b ER of the vector space R³. c=1} (i) The subset U (ii) The subset V = = a+b {(2+2) b 2a + 3b a {0) C € R³: a+b+c=1 of the vector space R³. (iii) The set D of matrices of determinant 0, in the vector space M2×2 (R) of all real 2×2 matrices. (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 coefficients in R. (v) The set Z of all sequences which are eventually zero, Z = {v = (vo, V₁, V2, ...) € F∞: there is n such that vi in the vector space F of infinite sequences v = (vo, V₁, V2, ...) with v; ¤ F (F any field). 0 for all i≥n},
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