Let Rz, be a linear space with all polynomials p(x)= ar² + bx+c(a,b,c R) whose degree is no more than 2. Let L be the operator on R[r], defined by Lar² + bx + c)= (a + 3b+6c)² + (a-3b-4c) x + (a+b+c) (1) Find the matrix A representing L with respect to basis [1,2r, 3x²]; (2) Find the matrix B representing with respect to basis [1-2r+3r²,-1+2r, - 1+32²]; (3) Is the matrix A diagonalizable? Why?
Let Rz, be a linear space with all polynomials p(x)= ar² + bx+c(a,b,c R) whose degree is no more than 2. Let L be the operator on R[r], defined by Lar² + bx + c)= (a + 3b+6c)² + (a-3b-4c) x + (a+b+c) (1) Find the matrix A representing L with respect to basis [1,2r, 3x²]; (2) Find the matrix B representing with respect to basis [1-2r+3r²,-1+2r, - 1+32²]; (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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![Let Rz, be a linear space with all polynomials p(z) = ax² + bx+c(a,b,c € R) whose degree
is no more than 2. Let L be the operator on R[r], defined by
4
Lar²+ bx+c)=(a + 3b+6c)x²+(-a-3b-4c)x+(a+b+
3
(1) Find the matrix A representing with respect to basis [1, 2, 3x²];
2
3
(2) Find the matrix B representing with respect to basis (1-2r+32²,-1+ 2r, -1+3z²);
(3) Is the matrix A diagonalizable? Why?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F31ca2837-4270-45c6-8f74-bc32489909cd%2F77507953-74d3-4d7c-8e05-86eef3340a48%2Fyiaaa5e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let Rz, be a linear space with all polynomials p(z) = ax² + bx+c(a,b,c € R) whose degree
is no more than 2. Let L be the operator on R[r], defined by
4
Lar²+ bx+c)=(a + 3b+6c)x²+(-a-3b-4c)x+(a+b+
3
(1) Find the matrix A representing with respect to basis [1, 2, 3x²];
2
3
(2) Find the matrix B representing with respect to basis (1-2r+32²,-1+ 2r, -1+3z²);
(3) Is the matrix A diagonalizable? Why?
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