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Linear combination

In [mathematics] , linear combinations is a concept central to [linear algebra] and related fields of mathematics.
Most of this article deals with linear combinations in the context of a [vector space] over a [field] , with some generalizations given at the end of the article.
Definition
Suppose that K is a field and V is a vector space over K . As usual, we call elements of V [vector] s and call elements of K [scalars] .
If v 1,..., v n are vectors and a 1,..., a n are scalars, then the linear combination of those vectors with those scalars as coefficients is
:a_1 v_1 a_2 v_2 a_3 v_3 \cdots a_n v_n. \,
There is some ambiguity in the use of the term "linear combination" as to whether it refers to the expression or to its value. In most cases the value is meant, like in the assertion "the set of all linear combinations of v 1,..., v n always forms a subspace"; however one could also say "two different linear combinations can have the same value" in which case the expression must have been meant. The subtle difference between these uses is the essence of the notion of [linear dependence] : a family F of vectors is linearly independent precisely if any linear combination of the vectors in F (as value) is uniquely so (as expression). In any case, even when viewed as expressions, all that matters about a linear combination is the coefficient of each v i ; trivial modifications such as permuting the terms or adding terms with zero coefficient are not considered to give new linear combinations.

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