Proving Inner Product

Question

Suppose $A$ is an $n \times n$ matrix.

How do I show that the inner product;

<A$\vec{u}$,$\vec{v}$> = <$\vec{u}$,A$^T$$\vec{v}$>

Answer 1

Using the fact that the inner product $\langle u, v \rangle$ can be written as $u^Tv$, we have

$$\langle Au, v \rangle = (Au)^Tv = (u^TA^T)v = u^T(A^Tv) = \langle u, A^Tv \rangle$$.