# Technically, we can use matrices in non-orthonormal basis, right?

Let's say there is a non-orthonormal basis formed by two vectors $$|1\rangle$$ and $$|2\rangle$$. There's some linear transformation $$A$$ which gives $$A|1\rangle=p|1\rangle+q|2\rangle$$, and $$A|2\rangle=r|1\rangle+s|2\rangle$$.

Now couldn't we still express $$A$$ as a 2x2 matrix of $$(p,q,r,s)$$? And it's not just a notation thing. We could still do the matrix multiplication of the $$A$$ matrix and a vector $$|v\rangle$$ to get the correct answer. This is assuming $$|v\rangle$$ is also expressed as a column matrix in the same non-orthonormal basis.

One property that we lose is that the ith row jth column of $$A$$ is no longer the inner product $$\langle i|A|j\rangle$$. Is this why we only work with matrices in an orthonormal basis?

Yes. Once you fix a basis, you can always express a linear operator as a matrix relative to that basis. Orthonormality is not at all needed. It's just that sometimes, orthonormality may simplify the calculations for the matrix entries $$a_{ij}$$ (and sometimes it may not). In fact, many linear algebra books (eg Friedberg, or Axler, or Hoffman Kunze) do not even introduce the concept of inner products/orthogonality until much later on in the books (chapters 6,6,8 respectively).