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 = (b) V = R³, W = (c) V = M₂ (R), W = {2×2 matrices with integer entries}. M₂(R), W = { [a b] : a, b, c, d € R, a+b+c+d=0}. (e) V = M₂ (R), W = {M : det(M) = 0}. (d) V = {(x, y) : x, y ≤ R, x ≥ 0}. {(x, y, z) : x, y, z € R, z = x + 2y}.
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 = (b) V = R³, W = (c) V = M₂ (R), W = {2×2 matrices with integer entries}. M₂(R), W = { [a b] : a, b, c, d € R, a+b+c+d=0}. (e) V = M₂ (R), W = {M : det(M) = 0}. (d) V = {(x, y) : x, y ≤ R, x ≥ 0}. {(x, y, z) : x, y, z € R, z = x + 2y}.
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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Just solve d and e
![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%2F56a470aa-0cf0-474f-85ec-ae3434943ef9%2F77e63dc6-0f0f-4df6-8a46-bac9db7fbb3e%2F9puep4_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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