mathematics

Generally, a matrix can be represented as how it transforms the two standard basis vectors, $\begin{bmatrix} 1\\ 0 \end{bmatrix}$ and $\begin{bmatrix} 0\\ 1 \end{bmatrix}$.

For instance, one may consider the 90 degree clockwise rotation matrix, $\begin{bmatrix} 0 & -1\\ 1 & 0 \end{bmatrix}$ which moves the basis vector $\begin{bmatrix} 1\\ 0 \end{bmatrix}$ to a new basis , $\begin{bmatrix} 0\\ 1 \end{bmatrix}$ and the basis vector $\begin{bmatrix} 0\\ 1 \end{bmatrix}$ to another new basis, $\begin{bmatrix} -1\\ 0 \end{bmatrix}$. Due to how all vectors are formed in $R^2$, they can be represented as a linear combination of those basis vectors $\begin{bmatrix} 1\\ 0 \end{bmatrix}$ and $\begin{bmatrix} 0\\ 1 \end{bmatrix}$; furthermore, all matrices represent linear transformations, and, as such, take the form $A \in R^{m \times n}$, meaning the matrix A is a linear transformation from $R^n$ to $R^m$, (which, I may add, is horrible notation used so that m by n is rows by columns, meaning m is the height and n is the width), is such that

$A(ax+by) = aAx + bAy$ $\forall a, b \in R, x, y \in R^n$.

This is why the general formula, given to students never to be fundamentally understood, is

$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ $\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} =$ $\begin{bmatrix} ax_1+bx_2\\ cx_1+dx_2 \end{bmatrix} \forall x, y \in R^2.$

This is because any vector, in this case $\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$, is simply a linear combination of the standard basis vectors $\begin{bmatrix} 1\\ 0 \end{bmatrix}$ and $\begin{bmatrix} 0\\ 1 \end{bmatrix}$, being equal to $x_1 \cdot \begin{bmatrix} 1\\ 0 \end{bmatrix} + x_2 \cdot \begin{bmatrix} 0\\ 1 \end{bmatrix}$.

This also leads to the general formula for multiplying matrices together, where you take those same vectors from the second matrix and you move them to the basis vectors of the first matrix. For instance, $\begin{bmatrix} 1 & 2\\ 2 & 3 \end{bmatrix} \cdot \begin{bmatrix} 3 & 2\\ -4 & 3 \end{bmatrix}$ takes the first column in the second matrix, $\begin{bmatrix} 3\\ -4 \end{bmatrix}$, as the first basis vector, and the second column in the second matrix, $\begin{bmatrix} 2\\ 3 \end{bmatrix}$, as the second basis vector. This results in $\begin{bmatrix} 1 \cdot 3 + 2 \cdot -4\\ 2 \cdot 3 + 3 \cdot -4 \end{bmatrix} = \begin{bmatrix} -5\\ -6 \end{bmatrix}$ as the new first basis vector and $\begin{bmatrix} 1 \cdot 2 + 2 \cdot 3\\ 2 \cdot 2 + 3 \cdot 3 \end{bmatrix} = \begin{bmatrix} 8\\ 13 \end{bmatrix}$ as the new second basis vector, forming the matrix $\begin{bmatrix} -5 & 8\\ -6 & 13 \end{bmatrix}$. More generally, when multiplying $2\times2$ matrices, $\begin{bmatrix} a & b\\ c & d \end{bmatrix} \cdot \begin{bmatrix} e & f\\ g & h \end{bmatrix}$ moves the basis vector $\begin{bmatrix} e\\ g \end{bmatrix}$ to the new basis $\begin{bmatrix} ae+bg\\ ce+dg \end{bmatrix}$ and the basis vector $\begin{bmatrix} f\\ h \end{bmatrix}$ to the new basis $\begin{bmatrix} af+bh\\ cf+dh \end{bmatrix}$, resulting in the new $2\times2$ matrix $\begin{bmatrix} ae+bg & af+bh\\ ce+dg & cf+dh \end{bmatrix}$.