3. Let V be a finite-dimensional inner product space. Prove the following statements. (a) If S₁ and S₂ are disjoint subsets of V such that S₁ U S₂ is an orthogonal basis for V, then span(S₁) and span(S₂) are orthogonal complementary pair in V. (b) Conversely, if W₁ and W₂ are subspaces of V that form an orthogonal comple- mentary pair, and S, is an orthogonal basis for W₁, i = 1,2, then S₁ and S₂ are disjoint and S₁ U S₂ is an orthogonal basis for V.
3. Let V be a finite-dimensional inner product space. Prove the following statements. (a) If S₁ and S₂ are disjoint subsets of V such that S₁ U S₂ is an orthogonal basis for V, then span(S₁) and span(S₂) are orthogonal complementary pair in V. (b) Conversely, if W₁ and W₂ are subspaces of V that form an orthogonal comple- mentary pair, and S, is an orthogonal basis for W₁, i = 1,2, then S₁ and S₂ are disjoint and S₁ U S₂ is an orthogonal basis for V.
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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