Difference between revisions of "Linear combination"
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==Definition== | ==Definition== | ||
Let {{M|(V,\mathcal{K})}} be a [[vector space]] and let {{M|v_1,v_2,\ldots,v_n\in V}} be given. A ''linear combination'' of {{M|v_1,\ldots,v_n}}'' is any vector of the form{{rFAVIDMH}}: | Let {{M|(V,\mathcal{K})}} be a [[vector space]] and let {{M|v_1,v_2,\ldots,v_n\in V}} be given. A ''linear combination'' of {{M|v_1,\ldots,v_n}}'' is any vector of the form{{rFAVIDMH}}: | ||
Latest revision as of 07:37, 29 July 2016
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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
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