[University Linear Algebra] Does this proof lose generality?

In one of the examples provided at my uni's reference book, there's a proof for the following statement: "Given a vector space K which is either the real or complex vector space, two square matrices A and B are similar if and only if A and B are associated matrices of one linear operator over K".
Since it's bidirectional, they prove both directions separately, but they start the left-to-right part by considering a linear operator T(x_1, x_2, ....x_n) = A*(x_1,x_2, ...x_n). Then they deduce that A is the associated matrix of T with the standard basis of K, and continue the proof from there.
Shouldn't a proper proof be independent of whether A has that property or not? How does that proof supposedly work if neither A nor B are associated matrices of T with the standard basis of K?