Prove that $Au\cdot v = u\cdot A^Tv$ with standard inner product

Question

Let A be an $n$ x $n$ matrix and let $u$,$v$ $\in$ $\mathbb{F^n}$. Prove that $$Au\cdot v = u\cdot A^Tv$$

when <,> is the standard inner product. how can I prove it ?

Answer 1

We have that $$Au \cdot v=\sum_{i,j} A_{ij}u_j v_i $$ Can you express the other one?