Vector Spaces

Vector spaces over $\FF[2]$. The n-dimensional space, $\FF[2]^n$, over $\FF[2]$ is the set of vectors with n elements in $\FF[2]$.
$\FF[2]^n = \left\{q \left( \begin{array}{c} a_1\\ a_2\\ \vdots\\ a_n \end{array}\right) = \vec a \vert a_i \epsilon \FF[2] \right\}$

We can add vectors as follows:
$\vec a +\vec b$ = $$\left( \begin{array}{c} a_1\\ \vdots\\ a_n \end{array} \righ... ...+b_1\\ \vdots\\ a_n+b_n \end{array} \right) \epsilon \FF[2]^n$$

We can scalar multiply by $\lambda\epsilon\FF[2]$ as:
$$\lambda \vec a = \lambda \left( \begin{array}{c} a_1\\ \vdo... ...1\\ \vdots\\ \lambda a_n \end{array} \right) \epsilon \FF[2]^n$$
Make note that $\FF[2]^n$ has $2^n$ elements, and is not equal to $\FF[2^n]$ There is nothing special about $\FF[2]$, $\FF[8], \FF[27], \FF[5]$ etc., and the rules over those fields are no different. The different between $\FF[2]^n$ and $\FF[2^n]$ is described in the finite field sections. $\FF[2]^2 = \{ {0 \choose 0}, {0 \choose 1} , {1 \choose 0} , {1 \choose 1} \}$ has four elements.