A linear combination is an expression formed by multiplying each of a collection of objects by scalars and adding the results. In elementary settings the objects are numbers or vectors and the scalars are real or complex numbers; in more general algebraic contexts the scalars come from a field or ring. The idea captures how one element can be built from others using only scaling and addition.
Formal idea and notation
Given elements v1, v2, …, vn and scalars a1, a2, …, an, the expression a1 v1 + a2 v2 + … + an vn is a linear combination of the vi with coefficients ai. If a combination equals a particular target element w, we say w is a linear combination of the vi. The set of all such combinations is called the span of {v1, …, vn}.
Common examples
- In R^2, 3(1,0) + 2(0,1) = (3,2) is a linear combination of the standard basis vectors.
- Polynomials: 2 + 5x − x^2 is a linear combination of 1, x, x^2 with coefficients 2, 5, −1.
- Functions: a f + b g denotes the pointwise combination a·f(x) + b·g(x).
Properties and distinctions
Linear combinations underpin notions of linear dependence and independence: a set is dependent if some nontrivial linear combination gives the zero element. The span gives the smallest linear subspace containing a set. In modules over rings, similar definitions hold but some properties differ because scalars need not form a field.
History and applications
The concept evolved with the development of linear algebra and vector spaces in the 19th and early 20th centuries. Linear combinations are fundamental in solving linear systems, expressing solutions with matrices, defining bases and coordinates, and appear throughout applied mathematics, physics, data science, and engineering wherever linear models and superposition apply.