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Mathematics-Online lexicon:

Linear Combination

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Given vectors $ v_1, v_2,\dots,v_m$ in a $ K$-vector space and given $ \lambda_i \in K$. A sum of the form

$\displaystyle \lambda_1 v_1 + \lambda_2 v_2 +\dots + \lambda_m v_m = \sum_{i=1}^m \lambda_i v_i $

is called a linear combination of the vectors $ v_i$.

More generally:

Let $ W \subset V$. A liner combination from $ W$ is a linear combination of a finite number of vectors from $ W$, that is, it is a finite sum of the form

$\displaystyle \lambda_1 w_1 + \lambda_2 w_2 + \dots + \lambda_k w_k = \sum_{i=1}^k \lambda_i w_i \, , $

where $ w_i \in W$ and $ \lambda_i \in K$ für $ 1 \leq i \leq k$.

(Authors: App/Burkhardt/Kimmerle)
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  automatically generated 2/10/2005