Matrix representations of linear transformations

[narfarnst]

Homework Statement

Let g(x)=3+x and T(f(x))=f'(x)g(x)+2f(x), and U(a+bx+cx²)=(a+b,c,a-b). So T:P₂(R)-->P₂(R) and U:P₂(R)-->R³.
And let B and y be the standard ordered bases for P₂ and R³ respectively.
Compute the matrix representation of U (denoted ^y_B) and T ([T]^y_B) and their composition UT.

Homework Equations

None, really.

The Attempt at a Solution

So I get what to do here, I'm just a little hung up with the polynomials.
First, you use the transformations given and the standard bases, and transform the standard bases. B={1,x,x²} and y={(1,0,0), (0,1,0), (0,0,1)}.
So U(1)=(1,0,1), U(x)=(1,0,-1), and U(x²)=(0,1,0).
And T(1)=2, T(x)=3+3x, and T(x²)=6x+4x².

Now, I'm confused as to how I right the actual matrix of transformation for these. I know what when you're using just numbers (T:R³-->R² for example), you write your transformed B basis in terms of coefficients of your y basis. But I'm not sure how to do that here.

 

[kurt.physics]

Do I write T(1),T(x), and T(x2) in terms of coefficients of y? Or am I supposed to do this for U?Any help on how to go about this would be really appreciated. Thanks!