Changing Bases

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 = ΣAⁱ_jb^j
bⁱ = ΣBⁱ_ja^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 xⁱ expresses some vector v in terms of basis a:
v = Σ_ixⁱaⁱ = Σ_ixⁱ(Σ_jAⁱ_jb^j) = Σ_j(Σ_iAⁱ_jxⁱ)b^j = Σ_jx'^jb^j.
where x'ⁱ = Σ_jA^j_ix^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 aⁱ, there is a matrix C_ij such that
Cv = C(Σ_ixⁱaⁱ) = Σ_jC_ij(Σ_ixⁱaⁱ)