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
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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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.
Transcribed Image Text: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.
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