Example 1.2 The set of rational numbers Q and the set of real numbers R with the usual addition and multiplication are fields. The set of natural number N or integers Z is not a field. Let Z/2Z denote the set consisting of two elements {a, b} with the addition defined by a + a = a, a + b = b + a = b, b + b = a and with the multiplication defined by a a = a, a b = b a = a, b b = b. Then Z/2Z is a field.

Recall the definition of the field Z/2Z = {a, b} in Example 1.2. Let V be the set of all matrices x1 x2 x3 x4 M2(Z/2Z) such that there is an even number of i for which xi = a. Prove that V is a subspace of M2(Z/2Z).