Matrix Multiplication by Columns and by Rows

Theorem

Let

• $A={\left[\begin{array}{ccc}{\mathbf{a}}_1& \dots & {\mathbf{a}}_m\end{array}\right]}^{\prime }$
• $B=\left[\begin{array}{ccc}{\mathbf{b}}_1& \dots & {\mathbf{b}}_n\end{array}\right]$

Then

• $AB=A\left[\begin{array}{ccc}{\mathbf{b}}_1& \dots & {\mathbf{b}}_n\end{array}\right]=\left[\begin{array}{ccc}A{\mathbf{b}}_1& \dots & A{\mathbf{b}}_n\end{array}\right]$
• $AB=\left[\begin{array}{c}{\mathbf{a}}_1\\ ⋮\\ {\mathbf{a}}_n\end{array}\right]B=\left[\begin{array}{c}{\mathbf{a}}_{1}B\\ ⋮\\ {\mathbf{a}}_{n}B\end{array}\right]$