4. Let P2 {ao +ait+azt | a0, a1, a2 E R}. That is, P2 is the linear space of all polynomials of degree less than or equal to two. Let L: P P2 be the linear transformation defined by L(f(t)) = f(0) + f(3)t. For example, L(1+2t+3t2) (1 + 2(0) + 3(0)2) + (1+2(3) +3(3)²) t = 1+34t. Similarly, L(1) = 1+t and L(t) = 3t. %3D (a) Find L(t2) = (b) Assume that L is a linear transformation and that A = {1,t, t2} is the standard basis for P2. Give the definition of [La and then find it. (c) The set C = {1-t2,1 + 2,1+t+ t2} is another basis for P2. Find the change of basis matrix P such that (L]e = P '(L]aP.
4. Let P2 {ao +ait+azt | a0, a1, a2 E R}. That is, P2 is the linear space of all polynomials of degree less than or equal to two. Let L: P P2 be the linear transformation defined by L(f(t)) = f(0) + f(3)t. For example, L(1+2t+3t2) (1 + 2(0) + 3(0)2) + (1+2(3) +3(3)²) t = 1+34t. Similarly, L(1) = 1+t and L(t) = 3t. %3D (a) Find L(t2) = (b) Assume that L is a linear transformation and that A = {1,t, t2} is the standard basis for P2. Give the definition of [La and then find it. (c) The set C = {1-t2,1 + 2,1+t+ t2} is another basis for P2. Find the change of basis matrix P such that (L]e = P '(L]aP.
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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![4. Let P2= {ao +ait+azt | ao, a1, a2 E R}. That is, P2 is the linear space of all polynomials of degree less than or equal
to two. Let L: P2 P2 be the linear transformation defined by
L(f(t)) = f(0) + S(3)t.
For example, L(1+21 +3t) = (1+ 2(0) +3(0)2) + (1 + 2(3) + 3(3)) t =1+34t. Similarly, L(1) = 1+t and L(t) = 3t.
(a) Find L(t2) =
(b) Assume that L is a linear transformation and that A = {1,t, t2 is the standard basis for P2. Give the definition
of La and then find it.
%3D
(c) The set C = {1 – t2, 1 + t2, 1 +t + t2} is another basis for P2. Find the change of basis matrix P such that
(L]e = P '(L]aP.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d940ce8-cba2-4a95-af25-aae0739ca5aa%2Fc6880f99-650f-404b-b5e3-80701119cd60%2Fqjryym_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. Let P2= {ao +ait+azt | ao, a1, a2 E R}. That is, P2 is the linear space of all polynomials of degree less than or equal
to two. Let L: P2 P2 be the linear transformation defined by
L(f(t)) = f(0) + S(3)t.
For example, L(1+21 +3t) = (1+ 2(0) +3(0)2) + (1 + 2(3) + 3(3)) t =1+34t. Similarly, L(1) = 1+t and L(t) = 3t.
(a) Find L(t2) =
(b) Assume that L is a linear transformation and that A = {1,t, t2 is the standard basis for P2. Give the definition
of La and then find it.
%3D
(c) The set C = {1 – t2, 1 + t2, 1 +t + t2} is another basis for P2. Find the change of basis matrix P such that
(L]e = P '(L]aP.
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