1. Consider the vector space V = P₂ (R) with standard basis B = {1, x, x²} and the linear maps T:V→V, S: V→V, T(f) = ƒ(1) + ƒ(−1)x + ƒ(0)x², S(ax² + bx + c) = cx² + bx + a. (a) Find [T] and [S]. Then show that 1 1 1 -CAD 1 -1 1 0 01 [TS] (b) Compute [(TS)-¹]8. (c) What is (TS)−¹ (x² + x + 1) ?
1. Consider the vector space V = P₂ (R) with standard basis B = {1, x, x²} and the linear maps T:V→V, S: V→V, T(f) = ƒ(1) + ƒ(−1)x + ƒ(0)x², S(ax² + bx + c) = cx² + bx + a. (a) Find [T] and [S]. Then show that 1 1 1 -CAD 1 -1 1 0 01 [TS] (b) Compute [(TS)-¹]8. (c) What is (TS)−¹ (x² + x + 1) ?
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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![1. Consider the vector space V = P₂(R) with standard basis
B = {1, x, x²}
and the linear maps
T:V →V,
S: V→V,
T(f) = ƒ(1) + ƒ(−1)x + ƒ(0)x²,
S(ax² + bx + c) = cx² + bx + a.
(a) Find [T] and [S]. Then show that
В
[TS]
=
C
1 1 1
1 -1
1
0 0 1
(b) Compute [(TS)-¹].
(c) What is (TS)-¹ (x² + x + 1) ?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa574767a-1b28-48ab-9fd0-f3b0aa8802f7%2F196fc22b-f15c-4234-92d0-52d98049ebbb%2Fwu79eb8_processed.png&w=3840&q=75)
Transcribed Image Text:1. Consider the vector space V = P₂(R) with standard basis
B = {1, x, x²}
and the linear maps
T:V →V,
S: V→V,
T(f) = ƒ(1) + ƒ(−1)x + ƒ(0)x²,
S(ax² + bx + c) = cx² + bx + a.
(a) Find [T] and [S]. Then show that
В
[TS]
=
C
1 1 1
1 -1
1
0 0 1
(b) Compute [(TS)-¹].
(c) What is (TS)-¹ (x² + x + 1) ?
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