The problem: Let $L: R_4 \to R_3$ be defined by $$L([u_1, u_2 ,u_3 ,u_4]) = [u_1 ,(u_2+u_3), (u_3 + u_4)]$$ Let S and T be the natural bases for $R_4$ and $R_3$, respectively. Find the representation of L with respect to S and T.
My answer: The bases for S and T are just $I_4$ and $I_3$, respectively. Now transforming S I find easy because S has four vectors in it. However, what am I supposed to do with T? It only goes up to $u_3$, not $u_4$. Do I attach a row of zeros to make it have four vectors, or is this type of transformation impossible? By the way, my answer for the transformation of S is:
$$L(S) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ \end{bmatrix} $$