# Linear transformation

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• Feb 6th 2010, 07:17 PM
Dranalion
Linear transformation
Consider the linear transformation T : P2 -> P1 speci fied by T(a0 + a1x +
a2x^2) = (-3a0 + 14a1 - 6a2) + (a0 - 4a1 + 2a2)x.
(a) Write this an equivalent mapping from R^3 to R^2.
(b) Using your answer to (a) or otherwise, specify the standard matrix
representation of T.
(c) The set D = {-3 + x; 7 - 2x} is also a basis for P1. Write down the
transition matrix PS;D, where S1 is the standard basis for P1, and use
it to calculate the transition matrix PD;S.
(d) Calculate the matrix [T]D;S2.

Any help is much appreciated,

Dranalion
• Feb 6th 2010, 10:40 PM
Dranalion
Is anyone able to help?
• Feb 7th 2010, 03:00 AM
HallsofIvy
Quote:

Originally Posted by Dranalion
Consider the linear transformation T : P2 -> P1 speci fied by T(a0 + a1x +
a2x^2) = (-3a0 + 14a1 - 6a2) + (a0 - 4a1 + 2a2)x.

Have you tried anything yourself? You shouldn't ask others to help until you have tried yourself and then you should show what you have tried.

Quote:

(a) Write this an equivalent mapping from R^3 to R^2.
$a_0+ a_1x+ a_2x^2$ is associated with $(a_0, a_1, a_2)$ and $b_0+ b_1x$ with $(b_0, b_1)$. $(a_0, a_1, a_2)$ is mapped to $(-3a_0+ 14a_1- 6a_2, a_0- 4a_1+ 2a_2)$

Quote:

(b) Using your answer to (a) or otherwise, specify the standard matrix
representation of T.
There are several different ways to do this. I don't know which you have been taught. The simplest is to apply the transformation to (1, 0, 0), (0, 1, 0), and (0, 0, 1) in turn. The results will be the three columns of the matrix.

[quote](c) The set D = {-3 + x; 7 - 2x} is also a basis for P1. Write down the
transition matrix PS;D, where S1 is the standard basis for P1, and use
it to calculate the transition matrix PD;S.[quote]
The transition matrix changes each vector v, written as (a, b) with v= a+ bx, in the standard basis, into (p, q) where v= p(-3+x)+ v(7- 2x). Find what (a, b) is changed into and that will give the columns for the transition matrix.

Quote:

(d) Calculate the matrix [T]D;S2.
This is now just matrix multiplication.

Quote:

Any help is much appreciated,

Dranalion