Linear Transformations: Finding Matrix with Standard Basis

[baddin]

1. Give information
Let T: P3 ---> P3 be the linear transformation described by:
T(p(x))=p(x+1)+p(2-x).
Find the matrix of T with respect to the standard basis b {1,x,x^2,x^3}.

The Attempt at a Solution

I found the transformations on the standard basis b:
T(1) = 2
T(x) = 3
T(x^2) = 2x^2 -2x +5
T(x^3) = 9x^2 - 9x + 9
I am confused on what to do next...

 

[Dick]

1. Give information
Let T: P3 ---> P3 be the linear transformation described by:
T(p(x))=p(x+1)+p(2-x).
Find the matrix of T with respect to the standard basis b {1,x,x^2,x^3}.

The Attempt at a Solution

I found the transformations on the standard basis b:
T(1) = 2
T(x) = 3
T(x^2) = 2x^2 -2x +5
T(x^3) = 9x^2 - 9x + 9
I am confused on what to do next...

Write your functions so they look a little more like vectors, write a+bx+cx^2+dx^3 as the column vector [a,b,c,d]. So T(1)=2 becomes T([1,0,0,0])=[2,0,0,0]. Does that help?

 

[baddin]

Ok, then I should find T(1,0,0,0), T(0,1,0,0), T(0,0,1,0) and T(0,0,0,1) right?

 

[Dick]

Ok, then I should find T(1,0,0,0), T(0,1,0,0), T(0,0,1,0) and T(0,0,0,1) right?

Right. You really already did. Just write them as column vectors. Then those will be the columns of your matrix.

 

[baddin]

Okay thank you very much for your help =)