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
[hide]Definition
Let (V,K) be a vector space and let v1,v2,…,vn∈V be given. A linear combination of v1,…,vn is any vector of the form[1]:
Note: A linear combination is always a finite sum[1][Note 2]
See also
Notes
- Jump up ↑ Obviously, by definition of a vector space: (∑ni=1aivi)∈V
- Jump up ↑ 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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