In the vector space of real functions on [-, π], let the inner product be defined by ㅠ (f,g) = f( [ f(x) g(x) dx. Let W = span{1, x, x²} with the basis G = {1, x, x²} (a) Find an orthogonal basis B for W. (b) Write down a matrix P so that P[p]G [P]B for all quadratic polynomial pe W. Here [p] and [p]Å are coordinates with respect to the basis G and B respectively. Find [1 + x + x²] and [1+x+x²]ß. = (c) Define F: W → W by Fp(x) = xp'(x). Find the matrix representation of the F operator with respect to the orthogonal basis B.
In the vector space of real functions on [-, π], let the inner product be defined by ㅠ (f,g) = f( [ f(x) g(x) dx. Let W = span{1, x, x²} with the basis G = {1, x, x²} (a) Find an orthogonal basis B for W. (b) Write down a matrix P so that P[p]G [P]B for all quadratic polynomial pe W. Here [p] and [p]Å are coordinates with respect to the basis G and B respectively. Find [1 + x + x²] and [1+x+x²]ß. = (c) Define F: W → W by Fp(x) = xp'(x). Find the matrix representation of the F operator with respect to the orthogonal basis B.
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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![In the vector space of real functions on [-, π], let the inner product be defined by
(f;9) = [*_*f(x)g(x)dx.
-πT
Let W = span{1, x, x²} with the basis G = {1, x, x²}
(a) Find an orthogonal basis B for W.
(b) Write down a matrix P so that P[p]G
[PB for all quadratic polynomial
pЄ W. Here [p]G and [p]в are coordinates with respect to the basis G and B
respectively. Find [1 + x + x²]Ġ and [1 +x+x²]ß
=
(c) Define F: W → W by Fp(x) = xp'(x). Find the matrix representation of the
F operator with respect to the orthogonal basis B.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2F7fc0c0b9-36b6-44aa-a97e-c97c1cb54fb7%2Fq5xfjn9_processed.png&w=3840&q=75)
Transcribed Image Text:In the vector space of real functions on [-, π], let the inner product be defined by
(f;9) = [*_*f(x)g(x)dx.
-πT
Let W = span{1, x, x²} with the basis G = {1, x, x²}
(a) Find an orthogonal basis B for W.
(b) Write down a matrix P so that P[p]G
[PB for all quadratic polynomial
pЄ W. Here [p]G and [p]в are coordinates with respect to the basis G and B
respectively. Find [1 + x + x²]Ġ and [1 +x+x²]ß
=
(c) Define F: W → W by Fp(x) = xp'(x). Find the matrix representation of the
F operator with respect to the orthogonal basis B.
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