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 = {ve R²: v₁ + ₂ ≤0}; (b) F=R, V = {(2v1, 3, -2): vE R³, and v₁ + 2 + V3 = 0}; (c) F= R, V = {ve R²: vf - 2v1v2 + v² ≤0}; (d) F=R, V = {ve 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 = {ve R²: e¹1 = 2e"}; (i) F=R, V = {v € R³: e¹¹e¹2 = eº³}; F=R, V = {v ER (j) v converges as n → ∞0}; (k) F = R, V = {v € R: Un converges absolutely as n → ∞}; (1) F= R, V = {v € R: un converges conditionally as n → ∞0};
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 = {ve R²: v₁ + ₂ ≤0}; (b) F=R, V = {(2v1, 3, -2): vE R³, and v₁ + 2 + V3 = 0}; (c) F= R, V = {ve R²: vf - 2v1v2 + v² ≤0}; (d) F=R, V = {ve 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 = {ve R²: e¹1 = 2e"}; (i) F=R, V = {v € R³: e¹¹e¹2 = eº³}; F=R, V = {v ER (j) v converges as n → ∞0}; (k) F = R, V = {v € R: Un converges absolutely as n → ∞}; (1) F= R, V = {v € R: un converges conditionally as n → ∞0};
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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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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