A transformation $T: V\to W$ transform m-dimensional space $V$ to n-dimensional space $W$ is given by $n\times m$ matrix $M$, such that we map vector $x$ from $V$ to $W$ as follows:
$\mathbf{x}^i = \mathbf{M} \mathbf{x}$
My question is that what is $M$? is it the orthonormal basis of $W$.
$M$ is the matrix corresponding to the linear transformation $T$, expressed in some (possibly orthonormal) bases for $V$ and $W$.