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:

(1)
\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 vT by u to obtain the cross product such that:

(3)
\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:

(4)
\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}