Linear combination

From Maths
Jump to: navigation, search
Stub grade: B
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Flesh out with a bigger see also section. Find some more references and add some comments about why this is important (linear independence)


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


  1. Obviously, by definition of a vector space: [ilmath]\left(\sum_{i=1}^na_iv_i\right)\in V[/ilmath]
  2. 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.


  1. 1.0 1.1 Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha

Template:Vector spaces navbox

Template:Functional analysis navbox