find the change of basis matrix from the basis B = (1, t-3, (t-3)^2) to the standard basis U = (1, t, t^2)
The "change of basis matrix" maps the coefficients of vectors written in terms of the first basis to the coefficients of the same vectors written in the second basis.
Further, a simple way of finding the matrix corresponding to a linear transformation in terms of bases for the "domain" and "range" spaces is:
Apply the linear transformation to each of the basis vectors for the domain in turn. Write the result in terms of the range basis. The coefficients for that linear combination form the column of the matrix.
For example, 1 is just mapped into 1 since the first "basis vector" in each basis is the same: 1= 1(1)+ 0(t)+ 0(t^2) so the first column of the matrix is $\displaystyle \begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix}$.
t- 3 is mapped into -3(1)+ 1(t)+ 0(t^2) so the second column of the matrix is $\displaystyle \begin{bmatrix}-2 \\ 1 \\ 0\end{bmatrix}$.
Can you do the third column?