Matrix representation of T with Basis B?

[Fernando Revilla]

I quote un unsolved question posted in MHF on December 8th, 2012 by user bonfire09.

Problem is assuming the mapping T: P2---->P2 defined by T(a0+a1t+a2t2)=3a0+(5a0-2a1)t+(4a1+a2)t^2 is linear. Find the matrix representation of T relative to Basis B={1,t,t^2}.
The part that I am confused on is when I go plug in the basis values T(1),T(t),and T(t^2)? I don't know how to do it?

From my understanding T(a0+a1t+a2t2) can be written as a0T(1)+a1T(t)+a2T(t^2)
So to find T(1) its just a0T(1)+0T(t)+0T(t^2)=3a0+5a0t? Am i right?

 

[Fernando Revilla]

Really this is not a problem relative to change of basis. Take into account that according to a well-known theorem, to find the matrix of a linear map $$f:V\to W$$ with respect to basis $$B_V,B_W$$ of $$V,W$$ respectively, you only need to find the coordinates of the image of the elements of $$B_V$$ with respect to $$B_W$$ and then, to write these as columns.

In our case $$T$$ is an endomorphism and only one basis is mentioned, so $$B_V=B_W=B$$. We have:
$$T(1)=3+5t\\T(t)=-2t+4t^2\\T(t^2)=t^2$$
Hence, the corresponding matrix representation is:
$$[T]_B^B=\begin{bmatrix}{3}&{\;\;0}&{0}\\{5}&{-2}&{0}\\{0}&{\;\;4}&{1}\end{bmatrix}$$