2 In each of the following cases, determine whether the subset W is a subspace of the given vector space V. Justify your answers: check the two closure axioms, the zero axiom and the negative axiom, and say which if any fail. (a) V = R², W = {(x, y) : x, y ≤ R, x>0}. (b) V = R³, W = {(x, y, z) : x, y, z € R, z = x + 2y}. (c) V = M₂ (R), W = {2×2 matrices with integer entries}.
2 In each of the following cases, determine whether the subset W is a subspace of the given vector space V. Justify your answers: check the two closure axioms, the zero axiom and the negative axiom, and say which if any fail. (a) V = R², W = {(x, y) : x, y ≤ R, x>0}. (b) V = R³, W = {(x, y, z) : x, y, z € R, z = x + 2y}. (c) V = M₂ (R), W = {2×2 matrices with integer entries}.
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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![2 In each of the following cases, determine whether the subset W is a subspace of the given vector space V.
Justify your answers: check the two closure axioms, the zero axiom and the negative axiom, and say which if
any fail.
(a) V = R², W = {(x, y) : x, y ≤ R, x ≥ 0}.
(b) V = R³, W = {(x, y, z) : x, y, z = R, z = x + 2y}.
(c) V = M₂ (R), W = {2×2 matrices with integer entries}.
(d) V = M₂ (R), W = { [a b] : a, b, c, d € R, a+b+c+d = 0}.
(e) V = M₂ (R), W = {M : det(M) = 0}.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6d94b61e-9088-4b29-bb35-c4fc8a801d78%2F025e4076-c7b4-4c31-bed8-9a29f23f94ba%2Fwzlglss_processed.png&w=3840&q=75)
Transcribed Image Text:2 In each of the following cases, determine whether the subset W is a subspace of the given vector space V.
Justify your answers: check the two closure axioms, the zero axiom and the negative axiom, and say which if
any fail.
(a) V = R², W = {(x, y) : x, y ≤ R, x ≥ 0}.
(b) V = R³, W = {(x, y, z) : x, y, z = R, z = x + 2y}.
(c) V = M₂ (R), W = {2×2 matrices with integer entries}.
(d) V = M₂ (R), W = { [a b] : a, b, c, d € R, a+b+c+d = 0}.
(e) V = M₂ (R), W = {M : det(M) = 0}.
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