The theory of finite-dimensional vector spaces was created primarily in connection with one problem, and that is the simultaneous solution of a system of k linear equations in n indeterminates over a field F of the form $$ \begin{gathered} {a_{11}}{X_1} + ... + {a_{1n}}{X_n} = {b_1} \hfill \\ {a_{21}}{X_1} + ... + {a_{2n}}{X_n} = {b_2} \hfill \\ {a_{k1}}{X_1} + ... + {a_{kn}}{X_n} = bk \hfill \\ \end{gathered} $$ where the a ij and the b i are elements of F and the X j are indeterminates taking values in F. Such systems arise in many applications and also in many areas of mathematics.