Matrix Multiplication

#Math

$\displaystyle A=n\times p,\space B=p\times m,\space AB=n\times m​$

$$AB=\begin{bmatrix}

			A\vec v_1&A\vec v_2&\cdots&A\vec v_m
			\end{bmatrix},\space B=\begin{bmatrix}
			\vec v_1&\vec v_2&\cdots&\vec v_m
			\end{bmatrix},\space A\vec v_i=v_{i_1}A_{j1}+v_{i_2}A_{j2}+\ldots + v_{i_m}A_{jm}$$

$T:\mathbb{R}^m\rightarrow\mathbb{R}^n,\space A=n\times m$

• Not commuative $AB \ne BA$
• Associative $A(BC) = (AB)C$
• Has identity, $AI_p=I_nA=A$
• Distributive for matrix and scalar multiplication $A(B+C)=AB+AC ,\forall A\in\mathbb{M},\mathbb{R}$