Problem is assuming the mapping T: P_{2}---->P_{2}defined by T(a_{0}+a_{1}t+a_{2}t^{2})=3a_{0}+(5a_{0}-2a_{1})t+(4a_{1}+a_{2})t^2 is linear. Find the matrix representation of T relative to Basis B={1,t,t^2}.

The part that im 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(a_{0}+a_{1}t+a_{2}t2) can be written as a_{0}T(1)+a_{1}T(t)+a_{2}T(t^2)

So to find T(1) its just a_{0}T(1)+0T(t)+0T(t^2)=3a_{0}+5a_{0}t? Am i right?