Why is an orthonormal basis ordered

Question

In textbook an orthonormal basis is defined as an ordered basis that is orthonormal. Is there any specific reason why we define it as "ordered"?

Answer 1

Typically any basis is by definition ordered, and that's necessary for practical computation with matrix representations: Given a basis $(E_a)$ of a vector space $\Bbb V$, the matrix representation of a vector $$v = \sum v^a E_a \in \Bbb V$$ is $$\pmatrix{v^1\\\vdots\\v^n} ,$$ so reordering the basis permutes the entries of the matrix representation, and in particular the matrix representation depends on the order.

If you've encountered orientations of vector spaces, you know that any (ordered) basis $\mathcal B$ determines an orientation by declaring $\mathcal B$ to be an oriented $n$-tuple. Permuting the basis elements preserves the orientation if the permutation is even but reverses it if the permutation is odd, so the determined orientation depends on order, too.