Matrix-vector multiplication

Given two length-

$n$ vectors

$\overrightarrow{v}$ and

$\overrightarrow{w}$, we define the scalar product

$\overrightarrow{v}\overrightarrow{w}$ as the value obtained by multiplying each element of

$v$ with the corresponding element of

$w$ and summing up the products:

$$\overrightarrow{v}\overrightarrow{w}=\sum _{i=1}^n{v}_i{w}_i$$

Given an

$m\times n$ matrix

$M$ and a length-

$n$ vector

$\overrightarrow{v}$, we define the matrix-vector product

$M\overrightarrow{v}$ as the

$m$ dimensional vector

$\overrightarrow{w}$, whose

$i$-th element is the scalar product of the

$i$-th row vector of

$M$ with the vector

$\overrightarrow{v}$:

$$\overrightarrow{w}=M\overrightarrow{v}\text{means}{w}_i=\sum _j{m}_{ij}{v}_j.$$

For

$M\overrightarrow{v}$ to be defined, the number of columns of

$M$ must be equal to the length of

$\overrightarrow{v}$.