Let R[r], be a linear space with all polynomials p(x) = az²+bx+c (a,b,c ER) whose degree is no more than 2. Let L be the operator on R[z], defined by L(p(x)) = (a + 3b + 3c)x² + (-3a-5b-3c)r + (3a + 3b + c) (1) Find the matrix A representing L with respect to basis [1, 2, ²]; (2) Find the matrix B representing L with respect to basis [1-x+x²,-1+x,-1+z²]; (3) Is the matrix A diagonalizable? Why?
Let R[r], be a linear space with all polynomials p(x) = az²+bx+c (a,b,c ER) whose degree is no more than 2. Let L be the operator on R[z], defined by L(p(x)) = (a + 3b + 3c)x² + (-3a-5b-3c)r + (3a + 3b + c) (1) Find the matrix A representing L with respect to basis [1, 2, ²]; (2) Find the matrix B representing L with respect to basis [1-x+x²,-1+x,-1+z²]; (3) Is the matrix A diagonalizable? Why?
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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![p(x) = ax² +bx+c (a,b,c R) whose degree
Let R[r], be a linear space with all polynomials
is no more than 2. Let L be the operator on R[2] defined by
L(p(x)) = (a + 3b + 3c)x² + (-3a-5b-3c)r + (3a + 3b + c)
(1) Find the matrix A representing with respect to basis [1,1,2²];
(2) Find the matrix B representing L with respect to basis [1-x+r²,-1+r, - 1+²];
(3) Is the matrix A diagonalizable? Why?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3fa1b71a-f2b0-4cdd-bc05-8db498629bb3%2Feda27f4d-b888-4e7b-bbb9-028b25f164e5%2Frdlpye6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:p(x) = ax² +bx+c (a,b,c R) whose degree
Let R[r], be a linear space with all polynomials
is no more than 2. Let L be the operator on R[2] defined by
L(p(x)) = (a + 3b + 3c)x² + (-3a-5b-3c)r + (3a + 3b + c)
(1) Find the matrix A representing with respect to basis [1,1,2²];
(2) Find the matrix B representing L with respect to basis [1-x+r²,-1+r, - 1+²];
(3) Is the matrix A diagonalizable? Why?
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