Matrix Multiplication

• Basically an extension of the Matrix-Vector Product.

$$AB=[A{\vec{b}}_1...A{\vec{b}}_k]$$

• The $nth$ row of the resultant matrix is the dot products kept side by side.
• It has the same num of cols as the $B$.

Alternate way to think: row dot column

$$\begin{array}{r}A\in M_{m\times n}(\mathbb{R})\\ B\in M_{n\times k}(\mathbb{R})\\ \\ A=\left[\begin{array}{c}{\vec{r}}_1^T\\ .\\ .\\ {\vec{r}}_m^T\end{array}\right]\\ \\ B=[{\vec{b}}_1...{\vec{b}}_k]\\ \\ AB=\left[\begin{array}{c}{\vec{r}}_1^T{\vec{b}}_1...{\vec{r}}_1^T{\vec{b}}_k\\ ...\\ {\vec{r}}_m^T{\vec{b}}_1...{\vec{r}}_m^T{\vec{b}}_k\end{array}\right]\end{array}$$

The $ith$ entry is ${\vec{r}}_i\cdot {\vec{b}}_j$

[!Matrix Multplication] $AB\ne BA$ It doesn’t work that way.