Order of vectors in a basis when computing representative matrix of a linear transformation with respect to said base

Question

We know that

$[T]^B_E = \begin{bmatrix} T(b_1) & \dots & T(b_n) \end{bmatrix}^B_E$

My question is very simple - if I change the order of T(b1),...,T(bn) I will get a different matrix. Is that matrix still considered a representative matrix of the transformation with respect to B, or is it incorrect to do so?

Answer 1

The real problem is that people often say "matrix wih respect to a basis" when what they mean is "matrix wih respect to an ordered basis." Strictly speaking, there is no such thing as "the matrix wih respect to that basis" since re-ordering an ordered basis produces a different matrix.