# Factor Matrix in the Inner Product

## Theorem

Let $\mathbf u$ and $\mathbf v$ be $1 \times n$ column vectors.

Then:

$\innerprod {A \mathbf u} {\mathbf v} = \innerprod {\mathbf u} {A^\intercal \mathbf v}$

## Proof

 $\ds \innerprod {A \mathbf u} {\mathbf v}$ $=$ $\ds \paren {A \mathbf u}^\intercal \mathbf v$ Definition of Dot Product $\ds$ $=$ $\ds \mathbf u^\intercal A^\intercal \mathbf v$ Transpose of Matrix Product $\ds$ $=$ $\ds \innerprod {\mathbf u} {A^\intercal \mathbf v}$ Definition of Dot Product

$\blacksquare$