Matrix Multiplication as Column Combinations

Mental Model

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

$$A=\left[\begin{array}{cc}\uparrow & \uparrow \\ {\vec{a}}_0& {\vec{a}}_1\\ \downarrow & \downarrow \end{array}\right]B=\left[\begin{array}{cc}\uparrow & \uparrow \\ {\vec{b}}_0& {\vec{b}}_1\\ \downarrow & \downarrow \end{array}\right]$$

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$$A\ast B=\left[\begin{array}{cc}\uparrow & \uparrow \\ {\vec{a}}_0& {\vec{a}}_1\\ \downarrow & \downarrow \end{array}\right]\ast \left[\begin{array}{cc}\uparrow & \uparrow \\ {\vec{b}}_0& {\vec{b}}_1\\ \downarrow & \downarrow \end{array}\right]=\left[\begin{array}{cc}\uparrow & \uparrow \\ A\ast {\vec{b}}_0& A\ast {\vec{b}}_1\\ \downarrow & \downarrow \end{array}\right]$$

$A$ multiplies every column vector in $B$.

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How A multiplies every Column Vector

$$\left[\begin{array}{c}\uparrow \\ A\ast {\vec{b}}_0\\ \downarrow \end{array}\right]=\left[\begin{array}{cc}\uparrow & \uparrow \\ {\vec{a}}_0& {\vec{a}}_1\\ \downarrow & \downarrow \end{array}\right]\ast \left[\begin{array}{c}({\vec{b}}_0{)}_0\\ \\ ({\vec{b}}_0{)}_1\end{array}\right]=({\vec{b}}_0{)}_0\ast \left[\begin{array}{c}\uparrow \\ {\vec{a}}_0\\ \downarrow \end{array}\right]+({\vec{b}}_0{)}_1\ast \left[\begin{array}{c}\uparrow \\ {\vec{a}}_1\\ \downarrow \end{array}\right]$$ $$\left[\begin{array}{c}\uparrow \\ A\ast {\vec{b}}_1\\ \downarrow \end{array}\right]=\left[\begin{array}{cc}\uparrow & \uparrow \\ {\vec{a}}_0& {\vec{a}}_1\\ \downarrow & \downarrow \end{array}\right]\ast \left[\begin{array}{c}({\vec{b}}_1{)}_0\\ \\ ({\vec{b}}_1{)}_1\end{array}\right]=({\vec{b}}_1{)}_0\ast \left[\begin{array}{c}\uparrow \\ {\vec{a}}_0\\ \downarrow \end{array}\right]+({\vec{b}}_1{)}_1\ast \left[\begin{array}{c}\uparrow \\ {\vec{a}}_1\\ \downarrow \end{array}\right]$$

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