Linear Maps

We think of a linear code as a linear map from

$$\underbrace{\mathbb{F}_{2}^n}_{\textrm{unencoded words}} \rightarrow \underbrace{\mathbb{F}_{2}^m}_{\textrm{where the encoded words live}}$$

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

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

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 $G^t:\mathbb{F}_{2}^k\rightarrow\mathbb{F}_{2}^n$ (or more generally $G^t:\mathbb{F}_{q}^k\rightarrow\mathbb{F}_{q}^n$) we have a set of codewords: Vector Space generated by $G^t(e_i)$

$$C = Im(G^t)$$

$$= \left\{ G^tw\vert w\epsilon\mathbb{F}_{2}^k\right\} \leq \mathbb{F}_{2}^n$$

$$= k<G^t(e_i)>$$

You may remember that a linear map L from one vector space V to W satisfies:

$$L:V \longrightarrow W$$

$$L(v_1+v_2) = L(v_1)+L(v_2)$$

$$L(\lambda v) = \lambda L(v)$$

$$L(0)=0$$

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:

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