Factor Matrix in the Inner Product

Theorem
$$=<\mathbf{u},A^T\mathbf{v}>$$, where $$\mathbf{u}$$ and $$\mathbf{v}$$ are both $$1\times n$$ column vectors.

Proof
By definition, the meaning of the above notation is $$<\mathbf{u},\mathbf{v}>=\mathbf{u}^T\mathbf{v}$$, so we get

$$=(A\mathbf{u})^T\mathbf{v}$$

From $$(AB)^T=B^TA^T$$, we get

$$=\mathbf{u}^TA^T\mathbf{v}$$

But this is, by definition, the same as $$<\mathbf{u},A^T\mathbf{v}>$$