Groups and Their Representations

Abstract

Any set of objects (called vectors) u, v,... that can be added and multiplied by real or complex numbers b, c,... will be called a real or complex vector space, provided the usual rules of addition and multiplication are satisfied:

$$ \begin{array}{*{20}{c}} {u + v = v + u} \\ {u + (v + w) = (u + v) + w} \\ {0 + u = u} \\ { - u + u = 0}\end{array} $$

$$ \begin{array}{*{20}{c}} {(u + v)c = uc + vc} \\ {u(b + c) = ub + uc} \\ {u(bc) = (ub)c} \\ {u1 = u}\end{array} $$