Our instructor wrote the following without doing a proof and I never got to ask him how he derived it.
Given an $n\times n$ Matrix $A$ and an orthonormal set of basis column vectors $\{\mathbf{v}_1, \mathbf{v}_2, ..., \mathbf{v}_n\}$ then $$ \begin{align*} \mathbf{v}_a^\intercal \mathbf{A} \, \mathbf{v}_b &= A_{ab} \end{align*} $$
I can see this for the standard basis but what about any general orthonormal basis? Any hints greatly appreciated!
The rule is basically saying that $V^TAV=A$ for any orthogonal matrix $V$. It is obviously false. If it were true then for any symmetric matrix $S$ we could perform diagonalization $D=V^TSV=S$ and conclude that any symmetric matrix were diagonal. The only matrix that satisfies this rule for all $V$ is a scalar multiple of identity (can be shown via Schur triangulation, for example).
P.S. Fix an ON basis $\{v_k\}$ for which it is true. Then taking $A=v_iv_j^T$ and applying the rule we get that the basis must be the standard one.