Linear transformations of matrices I'm currently trying to build an intuition for why Matrix multiplication works the way it does. However I can't seem to understand the concept where a Matric represents a linear transformation. What does this mean? How can a matrix represent a linear system of equations? Could you please explain it with as little mathematical jargon as possible.
 A: Good question. First we need a definition. We will work with $R^n$.
Linear Transformation: A linear transformation is a function $T : R^n\to R^m$ such that $$T(u+v)=T(u)+T(v)\text{ for all $u, v\in R^n$}$$ $$T(\lambda v)=\lambda T(v)\text{ for all $\lambda\in R$ and $v\in R^n$}$$
Let $v_1,\dots,v_n$ be a basis of $R^n$ and let $w_1,\dots,w_m$ be a basis of $R^m$. Now the matrix of a linear map $T : R^n \to R^m$ is the matrix whose columns consist of entries needed to write $Tv_k$ as a linear combination of $w_1,\dots,w_m$. In other words, the $k^{th}$ column of this matrix consists of the scalars in the following sum. $$Tv_k=A_{1, k}w_1+\dots+A_{m, k}w_m$$
Where $A_{n, m}$ are used to denote scalars in $R$. This definition is best understood by an example.
Consider the linear map $T : R^2\to R^2$ defined by $T(x, y)=(2x, 2y)$. With respect to the basis $(1, 0), (0, 1)$ of $R^2$, we see that $$T(1,0)=(2, 0)=2(1, 0)+0(0, 1)$$ $$T(0, 1)=(0, 2)=0(1, 0)+2(0, 1)$$ So, the matrix of $T$ with respect to the basis $(1, 0), (0, 1)$ $$\begin{bmatrix}2 & 0\\ 0 & 2 \end{bmatrix}$$ The first column consists of the scalars needed to write $T(1, 0)$ as a linear combination of $(1, 0), (0, 1)$ and the second column of this matrix consists of the scalars needed to write $T(0, 1)$ as a linear combination of $(1, 0), (0, 1)$. In this case, these scalars just so happen to be $T(1, 0)$ and $T(0, 1)$. But this is not always the case.

Now that we have a defintion for the matrix of a linear map. We want to know what is $M(ST)$. Where $M(ST)$ denotes the matrix of $ST$ and $S$ and $T$ are linear transformations. And $ST$ denotes the function composition of $S$ and $T$. To find this, we do a little computation.
Let $T : R^n\to R^m$ and $S : R^m\to R^h$ be linear transformations. Let $v_1,\dots,v_n$ be a basis of $R^n$, let $w_1,\dots,w_m$ be a basis of $R^m$, let $u_1,\dots,u_h$ be a basis of $R^h$. Let $A$ be the matrix of $T$ and let $C$ be the matrix of $S$. We want to know the scalars needed to write $(ST)v_k$ as a linear combination of $u_1,\dots,u_h$. We see that $$(ST)v_k=S(Tv_k)=S(\sum_{r=1}^m A_{r, k}w_r)$$ $$=\sum_{r=1}^mA_{r, k}S(w_r)=\sum_{r=1}^mA_{r, k}\sum_{j=1}^hC_{j, r}u_j$$ $$=\sum_{j=1}^h (\sum_{r=1}^m A_{r, k}C_{j, r})u_j= \sum_{j=1}^h (\sum_{r=1}^mC_{j, r} A_{r, k})u_j$$
We see that the scalars we need are $$\sum_{r=1}^mC_{j, r} A_{r, k} $$ Because we would like $M(ST)=M(S)M(T)$, we define matrix multiplication by the formula $$(CA)_{j, k}= \sum_{r=1}^mC_{j, r} A_{r, k} $$
Where $C$ is an $n\times m$ matrix and $A$ is an $m\times h$ matrix.
