Matrix Multiplication as Row Combinations

Mental Model

Given that A is a Rows matrix and B is a Rows matrix,

$$A=\left[\begin{array}{ccc}\leftarrow & {\vec{a}}_0& \to \\ \leftarrow & {\vec{a}}_1& \to \end{array}\right]B=\left[\begin{array}{ccc}\leftarrow & {\vec{b}}_0& \to \\ \leftarrow & {\vec{b}}_1& \to \end{array}\right]$$

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$$A\ast B=\left[\begin{array}{ccc}\leftarrow & {\vec{a}}_0& \to \\ \leftarrow & {\vec{a}}_1& \to \end{array}\right]\ast \left[\begin{array}{ccc}\leftarrow & {\vec{b}}_0& \to \\ \leftarrow & {\vec{b}}_1& \to \end{array}\right]=\left[\begin{array}{ccc}\leftarrow & {\vec{a}}_0\ast B& \to \\ \leftarrow & {\vec{a}}_1\ast B& \to \end{array}\right]$$

Every row vector in $A$ multiplies $B$.

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How every Row Vector multiplies B

$$\begin{array}{rl}\left[\begin{array}{ccc}\leftarrow & {\vec{a}}_0\ast B& \to \end{array}\right]& =\left[\begin{array}{cc}({\vec{a}}_0{)}_0& ({\vec{a}}_0{)}_1\end{array}\right]\ast \left[\begin{array}{ccc}\leftarrow & {\vec{b}}_0& \to \\ \leftarrow & {\vec{b}}_1& \to \end{array}\right]\\ & =({\vec{a}}_0{)}_0\ast \left[\begin{array}{ccc}\leftarrow & {\vec{b}}_0& \to \end{array}\right]+({\vec{a}}_0{)}_1\ast \left[\begin{array}{ccc}\leftarrow & {\vec{b}}_1& \to \end{array}\right]\end{array}$$ $$\begin{array}{rl}\left[\begin{array}{ccc}\leftarrow & {\vec{a}}_1\ast B& \to \end{array}\right]& =\left[\begin{array}{cc}({\vec{a}}_1{)}_0& ({\vec{a}}_1{)}_1\end{array}\right]\ast \left[\begin{array}{ccc}\leftarrow & {\vec{b}}_0& \to \\ \leftarrow & {\vec{b}}_1& \to \end{array}\right]\\ & =({\vec{a}}_1{)}_0\ast \left[\begin{array}{ccc}\leftarrow & {\vec{b}}_0& \to \end{array}\right]+({\vec{a}}_1{)}_1\ast \left[\begin{array}{ccc}\leftarrow & {\vec{b}}_1& \to \end{array}\right]\end{array}$$

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