Linear algebra - notation?

Niles

1. The problem statement, all variables and given/known data
I have a basis T and the elementary basis E in R^3 spanned by [e_1, e_2, e_3]. I am asked to find the matrix with respect to T and E.

1) Are they asking me to find the transition-matrix from T to E? If yes, then this is just the vectors that span T.

2) Or are they asking me to express T in terms of E, as in take L(T_1), L(T_2) and L(T_3) (L is my linear transformation) and express the result in terms of E - this result containts the columns for my matrix?

I am a little confused about this. I thought I could use B = U^(-1) * A * U, but apparently not?

I guess what I'm asking is - what is the difference between "the matrix with respect to the bases T and E" and "the transition matrix from T to E"?