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(d) F = R, V = {ve R²: v+v ≤ 0}; * (e) F = C, V = {ve C²: v² - 2v₁v2 + ² = 0}; (f) F= C, V = R²;

(d) F = R, V = {ve R²: v+v ≤ 0}; * (e) F = C, V = {ve C²: v² - 2v₁v2 + ² = 0}; (f) F= C, V = R²;

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.4: Linear Transformations
Problem 34EQ
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part D E F

In which of the following cases the set V together with the standard operation of addition and scalar multiplication forms a linear vector space? (If the set is a subset of a well-known vector space, then you only need to show that it is its subspace, i.e. verify that u + av € V for any u, v € V and a € F. If V is not a vector space, you need to find an example of u, v EV and a EF such that u + av V.) (a) F= R, V = {v € R²: v₁ + 0₂ ≤ 0}; (b) F=R, V = {(2v1, 3, -₂): vE R³, and v₁ + 2 + V3 = 0}; (c) F= R, V = {v € R²: v² - 2v1v2 + v² ≤0}; (d) F = R, V = {v € R²: v² + v² ≤ 0}; (e) F = C, V = {v € C²: v² - 2v₁v₂ + v² = 0}; (f) F = C, V = R²; (g) F = R, V = {v € R²: sin(v₁ + ₂) = 0}; (h) F= R, V = {v € R²: e¹¹ = 2e2}; (i) F= R, V = {v € R³: e¹e¹2 = e" }; (j) F= R, V = {ve Run converges as no}; (k) F= R, V = {ve Run converges absolutely as n → ∞}; (1) F= R, V = {vER: un converges conditionally as n → ∞0};
Transcribed Image Text:In which of the following cases the set V together with the standard operation of addition and scalar multiplication forms a linear vector space? (If the set is a subset of a well-known vector space, then you only need to show that it is its subspace, i.e. verify that u + av € V for any u, v € V and a € F. If V is not a vector space, you need to find an example of u, v EV and a EF such that u + av V.) (a) F= R, V = {v € R²: v₁ + 0₂ ≤ 0}; (b) F=R, V = {(2v1, 3, -₂): vE R³, and v₁ + 2 + V3 = 0}; (c) F= R, V = {v € R²: v² - 2v1v2 + v² ≤0}; (d) F = R, V = {v € R²: v² + v² ≤ 0}; (e) F = C, V = {v € C²: v² - 2v₁v₂ + v² = 0}; (f) F = C, V = R²; (g) F = R, V = {v € R²: sin(v₁ + ₂) = 0}; (h) F= R, V = {v € R²: e¹¹ = 2e2}; (i) F= R, V = {v € R³: e¹e¹2 = e" }; (j) F= R, V = {ve Run converges as no}; (k) F= R, V = {ve Run converges absolutely as n → ∞}; (1) F= R, V = {vER: un converges conditionally as n → ∞0};
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