Why is $\langle A^*x,y\rangle $ the same as $\langle A'^*y,x\rangle$

Question

If $A$ is a $3\times 3$-matrix, while $x$ and $y$ are a vector point with x,y,z. Why is the dot product of $\langle A^*x,y\rangle$ the same as the dot product of $\langle A'^*y,x\rangle$

($A'$ being the transposed matrix)

Answer 1

For column vectors $\mathbf{u},\mathbf{v}$, the scalar product is $$\langle \mathbf{u}, \mathbf{v} \rangle=\mathbf{u}^T \mathbf{v}. $$ Therefore $$\langle A \mathbf{x},\mathbf{y}\rangle = (A \mathbf{x})^T \mathbf{y}=\mathbf{x}^T A^T \mathbf{y}=\langle \mathbf{x},A^T \mathbf{y} \rangle = \langle A^T \mathbf{y}, \mathbf{x}\rangle. $$