A matrix times a column vector is a combination of the column of the matrix [21:10]
The matrix x col = col
This can be understand as applying linear transformation to a vector $[\textcolor{red}{a}, \textcolor{teal}{b}, \textcolor{green}{c}]^T$ to transform this vector from a vector space to another
$$ \left[ \begin{array}{c c c} . & . & . \\ . & . & . \\ . & . & . \end{array} \right] \left[ \begin{array}{c} \textcolor{red}{a} \\ \textcolor{teal}{b} \\ \textcolor{green}{c} \end{array} \right] \quad =\quad \textcolor{red}{a} \underset{\text{col 1}}{ \left[ \begin{array}{c} . \\ . \\ . \end{array} \right] } + \textcolor{teal}{b} \underset{\text{col 2}}{ \left[ \begin{array}{c} . \\ . \\ . \end{array} \right] } + \textcolor{green}{c} \underset{\text{col 3}}{ \left[ \begin{array}{c} . \\ . \\ . \end{array} \right] } \quad =\quad \left[ \begin{array}{c} . \\ . \\ . \end{array} \right] $$
A row vector times a matrix is a linear combination of rows of the matrix [22:15]
$$ \left[ \begin{array}{c c c} . & . & . \\ . & . & . \\ . & . & . \end{array} \right] \left[ \begin{array}{c} \textcolor{red}{a} \\ \textcolor{teal}{b} \\ \textcolor{green}{c} \end{array} \right] \quad =\quad \textcolor{red}{a} \underset{\text{col 1}}{ \left[ \begin{array}{c} . \\ . \\ . \end{array} \right] } + \textcolor{teal}{b} \underset{\text{col 2}}{ \left[ \begin{array}{c} . \\ . \\ . \end{array} \right] } + \textcolor{green}{c} \underset{\text{col 3}}{ \left[ \begin{array}{c} . \\ . \\ . \end{array} \right] } $$