Is anyone able to help?
Consider the linear transformation T : P2 -> P1 specified 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
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.
is associated with and with . is mapped to(a) Write this an equivalent mapping from R^3 to R^2.
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.(b) Using your answer to (a) or otherwise, specify the standard matrix
representation of T.
[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.
This is now just matrix multiplication.(d) Calculate the matrix [T]D;S2.
Any help is much appreciated,
Dranalion