When there are two competing sets of basis elements for a vector space it becomes necessary to convert expressions of vectors in one set, to expressions in the other, the expressed vector remaining the same. Likewise a linear transformation expressed as matrix is bound to one basis set and we will need to express the same transformation as another matrix in the other basis set. Sometimes such reasoning is referred to as the ‘alias’ perspective because it deals with two sorts of name for the same thing; in this case when two matrices depict the same transformation.

Our vector space has n dimensions. Each basis set is actually a numbered set of n vectors. The first set is ai and the second set is bi, i running from 1 thru n in each case. We will express each member of each set, in terms of the other set thus
ai = ΣAijbj
bi = ΣBijaj
In the above and in the following i, j and k each range independently over n values. As matrices, A and B are inverses of each other. If xi expresses some vector v in terms of basis a:
v = Σixiai = ΣixijAijbj) = ΣjiAijxi)bj = Σjx'jbj.
where x'i = ΣjAjixj and thus x'i expresses v in basis b.

Consider some linear transformation C from our space to itself. We consider a general v and u = Cv. For a particular basis ai, there is a matrix Cij such that
Cv = C(Σixiai) = ΣjCijixiai)