Matrix Multiplication

$A=n\times p,B=p\times m,AB=n\times m$

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 $A(B+C)=AB+AC \,\forall A\in\mathbb{M},\mathbb{R}$