Our vector space has n dimensions.
Each basis set is actually a numbered set of n vectors.
The first set is a^{i} and the second set is b^{i}, i running from 1 thru n in each case.
We will express each member of each set, in terms of the other set thus

a^{i} = ΣA^{i}_{j}b^{j}

b^{i} = ΣB^{i}_{j}a^{j}

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 x^{i} expresses some vector v in terms of basis a:

v = Σ_{i}x^{i}a^{i}
= Σ_{i}x^{i}(Σ_{j}A^{i}_{j}b^{j})
= Σ_{j}(Σ_{i}A^{i}_{j}x^{i})b^{j}
= Σ_{j}x'^{j}b^{j}.

where x'^{i} = Σ_{j}A^{j}_{i}x^{j} 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 a^{i}, there is a matrix C_{ij} such that

Cv = C(Σ_{i}x^{i}a^{i})
= Σ_{j}C_{ij}(Σ_{i}x^{i}a^{i})