User:Wongpak

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-===Linear Combination=== 
-Definition: Let (''u''<sub>i</sub>) = (''u''<sub>1</sub>, ''u''<sub>2</sub>, ..., ''u''<sub>n</sub>) be a sequence of vectors in V. A sum of the form<br> 
-::''a''<sub>i</sub> <math> \in </math> F, <math>\sum_{i=1}^n</math> ''a''<sub>i</sub>''u''<sub>i</sub> = ''a''<sub>1</sub>''u''<sub>1</sub> + ''a''<sub>2</sub>''u''<sub>2</sub>+ ... +''a''<sub>n</sub>''u''<sub>n</sub> 
-is called a "Linear Combination" of the ''u''<sub>i</sub>. 
===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}.

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