Master Code forms Vector Space

Theorem
Let $\map V {n, p}$ be a master code of length $n$ modulo $p$.

Then $\map V {n, p}$ forms a vector space over $\Z_p$ of $n$ dimensions.

Proof
Recall the vector space axioms:

First, the set of sequences $\tuple {x_1, x_2, \ldots, x_n}$, for $x_1, x_2, \ldots, x_n \in \Z_p$, has to be shown to fulfil the abelian group axioms.

This follows from:
 * Integers Modulo m under Addition form Cyclic Group

and:
 * Cyclic Group is Abelian.