I am horribly confused by the cluster of terminology and operations surrounding "change of basis" operations. Finding alternate references on this topic only seems to add to the confusion as there doesn't appear to be a consistent approach to defining and notating these operations. Perhaps someone will be able to clarify just one simple aspect of this which is as follows:
Let $u = \{u_1, \dots, u_n \}$ and and $w = \{w_1, \dots, w_n\}$ be bases for a vector space $V$. Then, necessarily, there exists a unique linear operator $T:V \rightarrow V$ such that $T(u_i) = w_i$. Now, the most natural thing in the world to call the matrix of this operator is the change of basis matrix from $u$ to $w$. Give this operator a vector in $u$ and it spits out a vector in $w$. Now, whether it is correct I don't know, but I've seen the matrix of this operator called the change of basis matrix from $w$ to $u$, reversing the target and source bases. This latter interpretation makes no sense because because it takes vectors in $u$ and produces vectors in $w$! I've seen this interpretation in more than one place so it can't just be a fluke. So...which is it?