Matrix Formula For Dot Product

Given two vectors **u** and **v** that exist in Euclidean n-Space, then we can write the dot product in terms of matrices. Consider matrix **u** and **v** that are column vectors:

\begin{bmatrix} u_1\\ u_2\\ ...\\ u_n \end{bmatrix}

And:

(2)\begin{bmatrix} v_1\\ v_2\\ ...\\ v_n \end{bmatrix}

If we take the transpose of of **v**, then we can multiply the transpose of **v**, denoted **v**T by **u** to obtain the cross product such that:

\begin{align} \vec{v}^T \vec{u} = \vec{u} \cdot \vec{v} \end{align}

# Matrix Multiplication on the Dot Product Proof

We will prove the following given a matrix A, and two column vectors **u**, and **v** that follows:

\begin{align} A\vec{u} \cdot \vec{v} = \vec{u} \cdot A^T \vec{v} \end{align}

We will prove this with matrix arithmetic:

(5)\begin{align} A\vec{u} \cdot \vec{v} = A \vec{v}^T \vec{u} \end{align}

(6)
\begin{align} A \vec{v}^T \vec{u} = (A^T \vec{v})^T \vec{u} \end{align}

(7)
\begin{align} (A^T \vec{v})^T \vec{u} = \vec{u} \cdot A^T \vec{v} \end{align}

Thus:

(8)\begin{align} A\vec{u} \cdot \vec{v} = \vec{u} \cdot A^T \vec{v} \end{align}