Prove that the following subset S of $\displaystyle Z_{2}~^5$ is a subspace of $\displaystyle Z_{2}~^5$. The subset S consists of all vectors with an even number of 1s, e.g. (1,0,1,0,0) and (0,0,0,0,0) are both in S while (1,0,1,0,1) is not in S.

Note: $\displaystyle Z_{2}$ = mod 2 or binary system