Let V be a vector space and W₁, W2 be subspaces of V. Then verify that W₁ + W₂ = {w₁ + W₂ : W₁ € W₁ and w₂ € W₂} is a subspace of V by verifying the following: (a) 0 € W₁ + W₂ (express the zero vector in the form v + w where v € W₁ and u E W₂) (b) Verify that if v € W₁ + W₂ and c E R is a scalar, then cv € W₁ + W₂. (c) Verify that if v, u € W₁ + W₂ then (v + w) ≤ W₁ + W₂.
Let V be a vector space and W₁, W2 be subspaces of V. Then verify that W₁ + W₂ = {w₁ + W₂ : W₁ € W₁ and w₂ € W₂} is a subspace of V by verifying the following: (a) 0 € W₁ + W₂ (express the zero vector in the form v + w where v € W₁ and u E W₂) (b) Verify that if v € W₁ + W₂ and c E R is a scalar, then cv € W₁ + W₂. (c) Verify that if v, u € W₁ + W₂ then (v + w) ≤ W₁ + W₂.
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:Let V be a vector space and W₁, W2 be subspaces of V. Then verify that
W₁ + W₂ = {w₁ +w2 : w₁ € W₁ and w₂ € W₂}
is a subspace of V by verifying the following:
(a) Ổ € W₁ + W₂ (express the zero vector in the form v + w where v € W₁ and u € W₂)
(b) Verify that if v € W₁ + W₂ and c E R is a scalar, then cv € W₁ + W₂.
(c) Verify that if v, ‚ u € W₁ + W₂ then (v + w) ≤ W₁ + W2.
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