Suppose \(T: V \to W\) is a linear transformation and \(B, B'\) are bases for \(V,W\), resp. Let \(B = \{ v_{1},\ldots,v_{n} \}\) and \(B' = \{ w_{1}, \ldots , w_{m} \}\). Then the *matrix of \(T\) with respect to the bases \(B,B'\) is the \(m \times n\) matrix \([T]_{B,B'}\) whose \(j\)th column is \([Tv_{j}]_{B'}\).

IOW, if \[ Tv_{j} = \sum_{i=1}^m a_{ij}w_{i}, \] then \([T]_{B,B'} = (a_{ij})\).

Note that \(T\) can be anything. It is not necessarily taking one base and converting it to the other. It simply represents what \(T\) does to the basis vectors of \(V\) and it takes values from \(V\) to \(W\).