Suppose $U$ is a vector space with basis $B=\set{\vectorlist{u}{n}}$ and the vector space $V$ contains the vectors $\vectorlist{v}{n}$ (which may not be distinct). Then there is a unique linear transformation, $\ltdefn{T}{U}{V}$, such that $\lt{T}{\vect{u}_i}=\vect{v}_i$, $1\leq i\leq n$.