Linear combination

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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).[1][2][3] 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.

Contents

Definition

A linear combination is the sum of some set of ordered pairs (here, vectors), each ordered pair weighed (multiplied) by some real number. For example, if you had the ordered pairs regarding age and height, and you had the set {(5,30), (10,40), (50,70)}, an instance of linear combination would be 4*(5,30)+6*(10,40)+3*(50,70) which would equal (230,570).

Suppose that K is a field (a set of real numbers) and V is a vector space over 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, like 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 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 vi; 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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