For each vector space V and subset W specified below, determine whether W is a subspace of V or not. Justify your answer. (a) V = {polynomials of order ≤ n with coefficients in R}, W = {polynomials of order exactly n with coefficients in R}. (b) V = R 3 , W = {(a, 1, b) : a, b ∈ R}. (c) V = R 2 , W = {(a, b) : a, b ≥ 0}. (d) V = {functions f : R → R}, W = {functions f : R → R such that d dx f(x) exists }, i.e. the set of differentiable functions.

For each vector space V and subset W specified below, determine whether W is a subspace of V or not. Justify your answer. (a) V = {polynomials of order ≤ n with coefficients in R}, W = {polynomials of order exactly n with coefficients in R}. (b) V = R 3 , W = {(a, 1, b) : a, b ∈ R}. (c) V = R 2 , W = {(a, b) : a, b ≥ 0}. (d) V = {functions f : R → R}, W = {functions f : R → R such that d dx f(x) exists }, i.e. the set of differentiable functions.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

For each vector space V and subset W specified below, determine whether W is a subspace of V or not. Justify your answer.

(a) V = {polynomials of order ≤ n with coefficients in R}, W = {polynomials of order exactly n with coefficients in R}.

(b) V = R 3 , W = {(a, 1, b) : a, b ∈ R}.

(d) V = {functions f : R → R}, W = {functions f : R → R such that d dx f(x) exists }, i.e. the set of differentiable functions.

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