Matrix Multiplication as Inner Products

What is the problem

You have a matrix $A^T$ that u view as a rows matrix, and want to multiply it by a columns matrix $B$.

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

You want to Compute $A^T\ast B$.

Before you begin

1. highlight each row vector in $A^T$ by a line that also shows its index.
   
2. highlight each column vector in $B$ by a line that also shows its index.
   

Their multiplication matrix is then constructed by computing the Inner product at every intersection.

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Connections

• Transposing a Matrix
• Matrix Multiplications as Outer Products