$A$ is a matrix in $\mathbb{R}^{m \times n}$ and $x \in \mathbb{R}^n$.
I want to prove that: $$\langle Ax, x \rangle = \langle x , A^T x \rangle $$
Having $A^T$ as the transpose of the matrix $A$.
Just realized I use this property so much, yet I don't known how to prove it.
We have that by definition
$$\langle Ax, x \rangle = (x^TA^T)x=x^T(A^Tx)=\langle x , A^T x \rangle$$
with $A\in \mathbb{R}^{n \times n}$, otherwise
$$\langle Ax, y \rangle = (x^TA^T)y=x^T(A^Ty)=\langle x , A^T y \rangle$$
with $y \in \mathbb{R}^m$.