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 ),

[ egin{equation*}

left.

egin{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}

ight},

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.