Linear combination

In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics.

Related Topics:
Mathematics - Linear algebra

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Most of this article deals with linear combinations in the context of a vector space over a field, with some generalisations given at the end of the article.

Related Topics:
Vector space - Field

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Definition   Suppose that K is a field and V is a vector space over K. As usual, we call elements of V vectors and call elements of K scalars.

Related Topics:
Vector - Scalars

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If v1,...,vn are vectors and a1,...,an are scalars, then the linear combination of those vectors with those scalars as coefficients is

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:a_1 v_1 + a_2 v_2 + a_3 v_3 + cdots + a_n v_n ,

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In a given situation, K and V may be specified explicitly, or they may be obvious from context.

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In that case, we often speak of a linear combination of the vectors v1,...,vn, with the coefficients unspecified (except that they must belong to K).

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Or, if S is a subset of V, we may speak of a linear combination of vectors in S, where both the coefficients and the vectors are unspecified, except that the vectors must belong to the set S (and the coefficients must belong to K).

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Finally, we may speak simply of a linear combination, where nothing is specified (except that the vectors must belong to V and the coefficients must belong to K).

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Note that by definition, a linear combination involves only of finitely many vectors (except as described in Generalisations below).

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However, the set S that the vectors are taken from (if one is mentioned) can still be infinite; each individual linear combination will only involve finitely many vectors.

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Also, there is no reason that n cannot be zero; in that case, we declare by convention that the result of the linear combination is the zero vector in V.

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