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Problem 4. Polynomials p₁(x) = 1, p₂(x) = x and p3(x) = r² form a basis for the vector space P3. Polynomials q₁(x) = 1, 2(x) = 1 + 2 and qa(z) = 1 +r+z² form another basis for P3- (i) Find the transition matrix from the ordered basis 91, 92, 93 to the ordered basis P₁, P2, P3. (ii) Find the transition matrix from the ordered basis P₁, P2, P3 to the ordered basis 91, 92, 93. (iii) Find coordinates of the polynomial r(x) = 2x² + 3x - 1 relative to the ordered basis 91, 92, 93-

Problem 4. Polynomials p₁(x) = 1, p₂(x) = x and p3(x) = r² form a basis for the vector space P3. Polynomials q₁(x) = 1, 2(x) = 1 + 2 and qa(z) = 1 +r+z² form another basis for P3- (i) Find the transition matrix from the ordered basis 91, 92, 93 to the ordered basis P₁, P2, P3. (ii) Find the transition matrix from the ordered basis P₁, P2, P3 to the ordered basis 91, 92, 93. (iii) Find coordinates of the polynomial r(x) = 2x² + 3x - 1 relative to the ordered basis 91, 92, 93-

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 4.** Polynomials \( p_1(x) = 1 \), \( p_2(x) = x \), and \( p_3(x) = x^2 \) form a basis for the vector space \(\mathcal{P}_3\). Polynomials \( q_1(x) = 1 \), \( q_2(x) = 1 + x \), and \( q_3(x) = 1 + x + x^2 \) form another basis for \(\mathcal{P}_3\). (i) Find the transition matrix from the ordered basis \( q_1, q_2, q_3 \) to the ordered basis \( p_1, p_2, p_3 \). (ii) Find the transition matrix from the ordered basis \( p_1, p_2, p_3 \) to the ordered basis \( q_1, q_2, q_3 \). (iii) Find coordinates of the polynomial \( r(x) = 2x^2 + 3x - 1 \) relative to the ordered basis \( q_1, q_2, q_3 \).
Transcribed Image Text:**Problem 4.** Polynomials \( p_1(x) = 1 \), \( p_2(x) = x \), and \( p_3(x) = x^2 \) form a basis for the vector space \(\mathcal{P}_3\). Polynomials \( q_1(x) = 1 \), \( q_2(x) = 1 + x \), and \( q_3(x) = 1 + x + x^2 \) form another basis for \(\mathcal{P}_3\). (i) Find the transition matrix from the ordered basis \( q_1, q_2, q_3 \) to the ordered basis \( p_1, p_2, p_3 \). (ii) Find the transition matrix from the ordered basis \( p_1, p_2, p_3 \) to the ordered basis \( q_1, q_2, q_3 \). (iii) Find coordinates of the polynomial \( r(x) = 2x^2 + 3x - 1 \) relative to the ordered basis \( q_1, q_2, q_3 \).
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