Let R[x] be a linear space with all polynomials p(x) = ax²+bx+c (a, b, c E R) whose degree is no more than 2. Let L be the operator on R[x], defined by 4 2 L(ax² + bx + c) = (a + 3b +6c)x² + (-=a −: 3b-4c)x+(a+b+c) -Ga 3 3 (1) Find the matrix A representing L with respect to basis [1,2x, 3x²]; (2) Find the matrix B representing L with respect to basis [1-2x+3x², -1 + 2x, -1+3x²]; (3) Is the matrix A diagonalizable? Why?
Let R[x] be a linear space with all polynomials p(x) = ax²+bx+c (a, b, c E R) whose degree is no more than 2. Let L be the operator on R[x], defined by 4 2 L(ax² + bx + c) = (a + 3b +6c)x² + (-=a −: 3b-4c)x+(a+b+c) -Ga 3 3 (1) Find the matrix A representing L with respect to basis [1,2x, 3x²]; (2) Find the matrix B representing L with respect to basis [1-2x+3x², -1 + 2x, -1+3x²]; (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 R[x] be a linear space with all polynomials p(x) = ax² + bx + c (a, b, c € R) whose degree
is no more than 2. Let L be the operator on R[x]3 defined by
4
2
L(ax² + bx + c) = (a + 3b +6c)x² + (-=a 3b4c)x+ ²a +
3
3
(1) Find the matrix A representing L with respect to basis [1,2x, 3x²];
a + b + c)
(2) Find the matrix B representing L with respect to basis [1-2x+3x2²,-1+2x, −1+3x²];
(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%2F2f240840-3076-4031-9d73-b8f8aae1bb0b%2F0xj5kv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let R[x] be a linear space with all polynomials p(x) = ax² + bx + c (a, b, c € R) whose degree
is no more than 2. Let L be the operator on R[x]3 defined by
4
2
L(ax² + bx + c) = (a + 3b +6c)x² + (-=a 3b4c)x+ ²a +
3
3
(1) Find the matrix A representing L with respect to basis [1,2x, 3x²];
a + b + c)
(2) Find the matrix B representing L with respect to basis [1-2x+3x2²,-1+2x, −1+3x²];
(3) Is the matrix A diagonalizable? Why?
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