Factor Matrix in the Inner Product

Theorem

 * $$\left \langle {A \mathbf{u},\mathbf{v}} \right \rangle = \left \langle {\mathbf{u},A^T\mathbf{v}} \right \rangle$$

where $$\mathbf{u}$$ and $$\mathbf{v}$$ are both $$1 \times n$$ column vectors.

Proof
By definition, the meaning of the above notation is $$\left \langle {\mathbf{u},\mathbf{v}} \right \rangle = \mathbf{u}^T \mathbf{v}$$, so we get:


 * $$\left \langle {A \mathbf{u},\mathbf{v}} \right \rangle = \left({A \mathbf{u}}\right)^T \mathbf{v}$$

From Transpose of Matrix Product, we have $$(AB)^T = B^T A^T$$, and so we get:


 * $$\left \langle {A \mathbf{u}, \mathbf{v}} \right \rangle = \mathbf{u}^T A^T \mathbf{v}$$

But this is, by definition, the same as $$\left \langle {\mathbf{u}, A^T \mathbf{v}} \right \rangle$$.