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Concept
Linear combination
Summary
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants). The concept of linear combinations is 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.
Let V be a vector space over the field K. As usual, we call elements of V vectors and call elements of K scalars.
If v1,...,vn are vectors and a1,...,an are scalars, then the linear combination of those vectors with those scalars as coefficients is
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 emphasized, as in the assertion "the set of all linear combinations of v1,...,vn always forms a subspace". However, one could also say "two different linear combinations can have the same value" in which case the reference is to the expression. 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 vi; trivial modifications such as permuting the terms or adding terms with zero coefficient do not produce distinct linear combinations.
In a given situation, K and V may be specified explicitly, or they may be obvious from context. 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). 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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