Real Dot Product

Definition: Real Dot Product

The dot product of two real column vectors $\vec{u}={\left[\begin{array}{ccc}{u}_1& \cdots & u_n\end{array}\right]}^{\mathsf{T}}\in {\mathbb{R}}^n$ and $\vec{v}={\left[\begin{array}{ccc}{v}_1& \cdots & v_n\end{array}\right]}^{\mathsf{T}}\in {\mathbb{R}}^n$ is defined as:

$$\vec{u}\cdot \vec{v}\stackrel{\text{def}}{=}\sum _{k=1}^n{u}_k{v}_k$$

NOTE

The dot product of two real column vectors is equivalent to the matrix product of the transpose of $\vec{u}$ with $\vec{v}$:

$$\vec{u}\cdot \vec{v}={\vec{u}}^{\mathsf{T}}\vec{v}$$