Let V R2[x] be the real vector space of polynomials of degree at most 2. Let {fo, f1, f2} with fo(x) = 1, f1 (x) = x, and f2(x) = x² be the standard basis of V. Define the inner product (, ·) on V by B = (f,8) = |, f(x)g(x) dx. (i) Use the Gram-Schmidt algorithm on B to find an orthogonal basis of V.
Let V R2[x] be the real vector space of polynomials of degree at most 2. Let {fo, f1, f2} with fo(x) = 1, f1 (x) = x, and f2(x) = x² be the standard basis of V. Define the inner product (, ·) on V by B = (f,8) = |, f(x)g(x) dx. (i) Use the Gram-Schmidt algorithm on B to find an orthogonal basis of 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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![Let V
R2[x] be the real vector space of polynomials of degree at most 2. Let
{fo, f1, f2} with fo(x) = 1, f1 (x) = x, and f2(x) = x² be the standard basis of V. Define the
inner product (', ·) on V by
B =
(f,8) = | f(x)g(x) dx.
(i)
Use the Gram-Schmidt algorithm on B to find an orthogonal basis of V.
(1) Find the best quadratic approximation to f(x) = x* on [0,1].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4c98b738-6ba7-4fd4-9d3b-c8f557e2a6c1%2F2e779f39-8bee-4747-8b10-f788b44c5a79%2F1fqspzf_processed.png&w=3840&q=75)
Transcribed Image Text:Let V
R2[x] be the real vector space of polynomials of degree at most 2. Let
{fo, f1, f2} with fo(x) = 1, f1 (x) = x, and f2(x) = x² be the standard basis of V. Define the
inner product (', ·) on V by
B =
(f,8) = | f(x)g(x) dx.
(i)
Use the Gram-Schmidt algorithm on B to find an orthogonal basis of V.
(1) Find the best quadratic approximation to f(x) = x* on [0,1].
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