Linear combination
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Flesh out with a bigger see also section. Find some more references and add some comments about why this is important (linear independence)
Contents
Definition
Let [ilmath](V,\mathcal{K})[/ilmath] be a vector space and let [ilmath]v_1,v_2,\ldots,v_n\in V[/ilmath] be given. A linear combination of [ilmath]v_1,\ldots,v_n[/ilmath] is any vector of the form^{[1]}:
 [math]\sum_{i=1}^na_iv_i[/math] for some scalars, [ilmath]a_1,a_2,\ldots,a_n\in\mathcal{K} [/ilmath]^{[Note 1]}.
Note: A linear combination is always a finite sum^{[1]}^{[Note 2]}
See also
Notes
 ↑ Obviously, by definition of a vector space: [ilmath]\left(\sum_{i=1}^na_iv_i\right)\in V[/ilmath]
 ↑ This is because in a vector space we only have binary operations, by induction we can apply the binary operation finitely many times, but not infinitely! More structure is needed to construct a limit.
References
