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Let p (x) = ao + a¡x + a2x² and q (x) = bo + b1x + b2x² be vectors in P2 with inner product < p, q >= aobo + a¡b1 + azb2 . Consider the polynomials in the set {2 + x², –1 + x + x² } 1. Show that the polynomials in the set do not form an orthonormal set. 2. Use the Gram-Schmidt orthonormalization process to form an orthonormal set from these polynomials. Keep the order, and show work for the orthogonalization, then normalization. 3. Do the polynomials in the orthonormal set (from question 2) form a basis for P2 ? Why or why not?

Let p (x) = ao + a¡x + a2x² and q (x) = bo + b1x + b2x² be vectors in P2 with inner product < p, q >= aobo + a¡b1 + azb2 . Consider the polynomials in the set {2 + x², –1 + x + x² } 1. Show that the polynomials in the set do not form an orthonormal set. 2. Use the Gram-Schmidt orthonormalization process to form an orthonormal set from these polynomials. Keep the order, and show work for the orthogonalization, then normalization. 3. Do the polynomials in the orthonormal set (from question 2) form a basis for P2 ? Why or why not?

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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Let \( p(x) = a_0 + a_1 x + a_2 x^2 \) and \( q(x) = b_0 + b_1 x + b_2 x^2 \) be vectors in \( P_2 \) with inner product \( \langle p, q \rangle = a_0 b_0 + a_1 b_1 + a_2 b_2 \). Consider the polynomials in the set \(\{ 2 + x^2, \, -1 + x + x^2 \}\) 1. Show that the polynomials in the set do not form an orthonormal set. 2. Use the Gram-Schmidt orthonormalization process to form an orthonormal set from these polynomials. Keep the order, and show work for the orthogonalization, then normalization. 3. Do the polynomials in the orthonormal set (from question 2) form a basis for \( P_2 \)? Why or why not?
Transcribed Image Text:Let \( p(x) = a_0 + a_1 x + a_2 x^2 \) and \( q(x) = b_0 + b_1 x + b_2 x^2 \) be vectors in \( P_2 \) with inner product \( \langle p, q \rangle = a_0 b_0 + a_1 b_1 + a_2 b_2 \). Consider the polynomials in the set \(\{ 2 + x^2, \, -1 + x + x^2 \}\) 1. Show that the polynomials in the set do not form an orthonormal set. 2. Use the Gram-Schmidt orthonormalization process to form an orthonormal set from these polynomials. Keep the order, and show work for the orthogonalization, then normalization. 3. Do the polynomials in the orthonormal set (from question 2) form a basis for \( P_2 \)? Why or why not?
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