As discussed in Chapter 1, one of the main goals of Linear Algebra is the characterization of solutions to a system of $$m$$ linear equations in $$n$$ unknowns $$x_1, \ldots, x_n$$,
$\begin{equation*} \left. \begin{array}{rl} a_{11} x_1 + \cdots + a_{1n} x_n &= b_1\\ \vdots \qquad \vdots \qquad & \quad \vdots\\ a_{m1} x_1 + \cdots + a_{mn} x_n &= b_m \end{array} \right\}, \end{equation*}$
where each of the coefficients $$a_{ij}$$ and $$b_i$$ is in $$\mathbb{F}$$. Linear maps and their properties give us insight into the characteristics of solutions to linear systems.