- I know if we have equal number of equation and unknown variables then we can always transform them into matrix form.
For example:
$$ \begin{aligned} 3x - y &= 0 \\ x - 2y &= 3 \end{aligned} $$
Above two can be transformed to
$$ \begin{bmatrix} 3 & -1 \\ 1 & -2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 3 \end{bmatrix} $$
But what about the system of equation in which number of unknowns are not same as number of equations. We can make those like:
$$ \begin{aligned} 2x + 3y + 4z - 3 &= 2 \\ 3y + 4z &= 7 \end{aligned} $$
System of finite linear equations, with finite number of variables, can be understand as linear transformation between some finite dimensional vector spaces.
Suppose we have $\mathbb{U}$ and $\mathbb{V}$ vector spaces over the same field and $\{x_1,\cdots,x_n \}$ and $\{y_1,\cdots,y_m \}$ are bases of $\mathbb{U}$ and $\mathbb{V}$ respectively. Corresponding matrix $A$ is, by definition, rectangular, $m$ by $n$, and defined by $$Ax_j=\sum\limits_{i}\alpha_{ij}y_i$$ where $\alpha_{ij}$ are scalars from mentioned field.
So, matrix theory can be extend to linear transformations, i.e. systems of linear equations, and vise versa.