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Basis

Basis $ \mathcal{B} = \{ b_1, b_2,
\hdots b_3 \}$ for a vector space V is a set of vectors so that any vector $ v_i$ in V can be written as a unique linear combination of these vectors. Ex. for the space $ \FF[2]^n$ (or more generally $ \FF[q]^n$) We always have a standard basis $ \mathcal{E}= \{e_1, \hdots e_n\}$ where the $ e_i \epsilon \FF[2]^n$ are:
$\displaystyle \vec e_1$ $\displaystyle =$ $\displaystyle \left( 1 \ 0 \ 0 \ 0 \dots\ 0 \ 0 \right)^{t}$  
$\displaystyle \vec e_2$ $\displaystyle =$ $\displaystyle \left( 0 \ 1 \ 0 \ 0\dots\ 0 \ 0 \right)^{t}$  
$\displaystyle \vec e_3$ $\displaystyle =$ $\displaystyle \left( 0 \ 0 \ 1 \ 0 \dots\ 0 \ 0 \right)^{t}$  
$\displaystyle \vdots$   $\displaystyle \vdots$  
$\displaystyle \vec e_n$ $\displaystyle =$ $\displaystyle \left( 0 \ 0 \ 0 \ 0 \dots\ 0 \ 1 \right)^{t}$  

This basis is a basis because for any $ \vec x =\left( \begin{array}{c}x_1\\
\vdots \\ x_n \end{array}\right) \epsilon \FF[q]^n$ we have: $ \vec x = x_1*\left(\begin{array}{c}1\\ 0\\ 0\\
\vdots\\ 0\end{array}\right)+...
... 0\\ 1\end{array}\right)= x_n \vec e_1 + x_2 \vec e_2 + \hdots + x_n
\vec e_n $ We will always use the standard basis (unless otherwise denoted). Knowing what happens to a basis (vector) given that pattern for what happens in every situation. $ L \left( \begin{array}{c}x_1\\ \vdots\\ x_n \end{array}\right) = L(x_n \vec e_1+
\hdots + x_n \vec e_n) = x_1L\vec e_1+ \hdots + x_nL\vec e_n$
next up previous contents
Next: Linear Maps Up: Vector Spaces Previous: Vector Spaces   Contents
Frederick Leitner 2004-05-12