Master Code forms Vector Space

From ProofWiki
Jump to: navigation, search

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:

The vector space axioms consist of the abelian group axioms:

\((V \, 0)\)   $:$   Closure Axiom      \(\displaystyle \forall \mathbf x, \mathbf y \in G:\) \(\displaystyle \mathbf x +_G \mathbf y \in G \)             
\((V \, 1)\)   $:$   Commutativity Axiom      \(\displaystyle \forall \mathbf x, \mathbf y \in G:\) \(\displaystyle \mathbf x +_G \mathbf y = \mathbf y +_G \mathbf x \)             
\((V \, 2)\)   $:$   Associativity Axiom      \(\displaystyle \forall \mathbf x, \mathbf y, \mathbf z \in G:\) \(\displaystyle \paren {\mathbf x +_G \mathbf y} +_G \mathbf z = \mathbf x +_G \paren {\mathbf y +_G \mathbf z} \)             
\((V \, 3)\)   $:$   Identity Axiom      \(\displaystyle \exists \mathbf 0 \in G: \forall \mathbf x \in G:\) \(\displaystyle \mathbf 0 +_G \mathbf x = \mathbf x = \mathbf x +_G \mathbf 0 \)             
\((V \, 4)\)   $:$   Inverse Axiom      \(\displaystyle \forall \mathbf x \in G: \exists \paren {-\mathbf x} \in G:\) \(\displaystyle \mathbf x +_G \paren {-\mathbf x} = \mathbf 0 \)             


together with the properties of a unitary module:

\((V \, 5)\)   $:$   Distributivity over Scalar Addition      \(\displaystyle \forall \lambda, \mu \in K: \forall \mathbf x \in G:\) \(\displaystyle \paren {\lambda + \mu} \circ \mathbf x = \lambda \circ \mathbf x +_G \mu \circ \mathbf x \)             
\((V \, 6)\)   $:$   Distributivity over Vector Addition      \(\displaystyle \forall \lambda \in K: \forall \mathbf x, \mathbf y \in G:\) \(\displaystyle \lambda \circ \paren {\mathbf x +_G \mathbf y} = \lambda \circ \mathbf x +_G \lambda \circ \mathbf y \)             
\((V \, 7)\)   $:$   Associativity with Scalar Multiplication      \(\displaystyle \forall \lambda, \mu \in K: \forall \mathbf x \in G:\) \(\displaystyle \lambda \circ \paren {\mu \circ \mathbf x} = \paren {\lambda \cdot \mu} \circ \mathbf x \)             
\((V \, 8)\)   $:$   Identity for Scalar Multiplication      \(\displaystyle \forall \mathbf x \in G:\) \(\displaystyle 1_K \circ \mathbf x = \mathbf x \)             


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.



Sources