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Linear Maps

We think of a linear code as a linear map from

$\displaystyle \underbrace{\FF[2]^n}_{\textrm{unencoded words}}
\rightarrow \underbrace{\FF[2]^m}_{\textrm{where the encoded words live}}$


(Here we denote the linear map by both $ L$ and $ G^t$. The later is used in the subsequent section on linear codes) Ex $ \FF[2]^7 \longrightarrow \FF[2]^8$

\begin{displaymath}\left( \begin{array}{c}
u_1\\
u_2\\
\vdots\\
u_7
\end...
...
\vdots\\
u_7\\
u_1+u_2+\hdots+u_7
\end{array}
\right) \end{displaymath}

The above is the linear map for adding a parity bit. The fact that this is a linear map means that every entry of the vector on the right is a linear function of the entries on the vectors on the left. For a linear code, which is a map $ G^t:\FF[2]^k\rightarrow\FF[2]^n$ (or more generally $ G^t:\FF[q]^k\rightarrow\FF[q]^n$) we have a set of codewords: Vector Space generated by $ G^t(e_i)$
$\displaystyle C$ $\displaystyle =$ $\displaystyle Im(G^t)$  
  $\displaystyle =$ $\displaystyle \left\{ G^tw\vert w\epsilon\FF[2]^k\right\} \leq \FF[2]^n$  
  $\displaystyle =$ $\displaystyle k<G^t(e_i)>$  

You may remember that a linear map L from one vector space V to W satisfies:
$\displaystyle L:V$ $\displaystyle \longrightarrow$ $\displaystyle W$  
$\displaystyle L(v_1+v_2)$ $\displaystyle =$ $\displaystyle L(v_1)+L(v_2)$  
$\displaystyle L(\lambda v)$ $\displaystyle =$ $\displaystyle \lambda L(v)$  
$\displaystyle L(0)=0$      

where 0= $ \left(\begin{array}{c}0\\ 0\\ \vdots \\ 0 \end{array}\right)$ is the zero vector. If we know what happens to a set of basis vectors $ L(\vec e_i)$ of a vector space V, then we know what happens to all vectors $ \vec v
\epsilon V$ because:

$\displaystyle L(\vec v)= L\left( \begin{array}{c}v_1\\ \vdots \\ v_n \end{array...
... e_1 + v_2 \vec e_2+\hdots+v_n \vec e_n)=v_1L(\vec
e_1)+\hdots+v_nL(\vec e_n) $


next up previous contents
Next: Sentences as Numbers Up: Vector Spaces Previous: Basis   Contents
Frederick Leitner 2004-05-12