How to know the arrangement of a basis?
e.g:
Let V be the span of sinx and cosx in C∞, T(f)= 3f + 2f'- f'', T: V → V,
1. Find B-matrix, 2. Is T an isomorphism?
Solving this question, I have to find the Basis first, but I don't know which basis is correct, because the arrangement do have impact on the answer.
Basis {sinx, cosx}, or Basis {cosx, sinx}
A basis is a set, and a set is an unordered list. So long as bases $\mathcal{B}$ and $\mathcal{C}$ have the same elements, the bases are the same.
I should add that the choice of basis order will affect the matrix, but only up to a permutation.