# User:Wongpak

(Difference between revisions)
 Revision as of 15:56, 29 September 2006Wongpak (Talk | contribs)Span← Previous diff Revision as of 18:06, 29 September 2006Wongpak (Talk | contribs) Next diff → Line 1: Line 1: - ===Linear Combination=== - Definition: Let (''u''i) = (''u''1, ''u''2, ..., ''u''n) be a sequence of vectors in V. A sum of the form
- ::''a''i $\in$ F, $\sum_{i=1}^n$ ''a''i''u''i = ''a''1''u''1 + ''a''2''u''2+ ... +''a''n''u''n - is called a "Linear Combination" of the ''u''i. ===Span=== ===Span===

## Revision as of 18:06, 29 September 2006

### Span

span(ui):= The set of all possible linear combinations of the ui's.

If $\mathcal{S} \subseteq$ V is any subset,

 span $\mathcal{S}$ := The set of all linear combination of vectors in $\mathcal{S}$ =$\left \{ \sum_{i=0}^n a_i u_i, a_i \in \mbox{F}, u_i \in \mathcal{S} \right \} \ni 0$

even if $\mathcal{S}$ is empty.

Theorem: For any $\mathcal{S} \subseteq$ V, span $\mathcal{S}$ is a subspace of V.

Proof:
1. 0 $\in$ span $\mathcal{S}$.
2. Let x $\in$ span $\mathcal{S}$, Let x $\in$ span $\mathcal{S}$, $\Rightarrow$ x = $\sum_{i=1}^n$ aiui, ui $\in \mathcal{S}$, y = $\sum_{i=1}^m$ bivi, vi $\in \mathcal{S}$. $\Rightarrow$ x+y = $\sum_{i=1}^n$ aiui + $\sum_{i=1}^m$ bivi = $\sum_{i=1}^{m+n}$ ciwi where ci=(a1, a2,...,an, b1, b2,...,bm) and wi=ci=(u1, u2,...,un, v1, v2,...,vm).
3. cx= c$\sum_{i=1}^n$ aiui=$\sum_{i=1}^n$ (cai)ui$\in$ span $\mathcal{S}$.