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Problem 6 (Orthogonal Complement). Consider the vector space P2 along with the inner product (f(x),9(x)) = [', f(x)g(x)dx (a) Find the orthogonal complement of W₁ = span(1+x). (b) Find the orthogonal complement of W₂ = span(1+x, x²). (c) Verify that W₂ ≤ W±. (d) The observation made in (c) is true in full generality: prove that if W₁ C W2 are two subspaces of an inner product space, then WC W.

Problem 6 (Orthogonal Complement). Consider the vector space P2 along with the inner product (f(x),9(x)) = [', f(x)g(x)dx (a) Find the orthogonal complement of W₁ = span(1+x). (b) Find the orthogonal complement of W₂ = span(1+x, x²). (c) Verify that W₂ ≤ W±. (d) The observation made in (c) is true in full generality: prove that if W₁ C W2 are two subspaces of an inner product space, then WC W.

Advanced Engineering Mathematics
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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Problem 6 (Orthogonal Complement). Consider the vector space P2 along with the inner product (f(x),9(x)) = [', f(x)g(x)dx (a) Find the orthogonal complement of W₁ = span(1+x). (b) Find the orthogonal complement of W₂ = span(1+x, x²). (c) Verify that W₂ ≤ W±. (d) The observation made in (c) is true in full generality: prove that if W₁ C W2 are two subspaces of an inner product space, then WC W.
Transcribed Image Text:Problem 6 (Orthogonal Complement). Consider the vector space P2 along with the inner product (f(x),9(x)) = [', f(x)g(x)dx (a) Find the orthogonal complement of W₁ = span(1+x). (b) Find the orthogonal complement of W₂ = span(1+x, x²). (c) Verify that W₂ ≤ W±. (d) The observation made in (c) is true in full generality: prove that if W₁ C W2 are two subspaces of an inner product space, then WC W.
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