Linear combination

A linear combination is a finite sum of objects multiplied by scalar coefficients. It describes how vectors and other elements of a vector space can be built from given elements.

Vector space: A set whose elements can be added and multiplied by scalars while obeying specific algebraic rules. Linear combinations are defined for elements of a vector space.
Linear independence: A property of vectors for which only the all-zero coefficients produce a zero linear combination. It tests whether a combination represents zero in a nontrivial way.
System of linear equations: A collection of equations in which each equation is linear in the unknowns. Solving a system often asks whether a target vector is a combination of given columns.
Affine span: The smallest affine subspace containing a given set of points. Unlike linear span, it need not contain the origin.
Scalar: An element of a field that multiplies vectors or other vector-space elements. Scalars supply the coefficients in a linear combination.
Basis (linear algebra): A linearly independent set of vectors whose span is the entire vector space. A basis gives each vector a unique linear-combination representation.
Column space: The span of a matrix's columns, consisting of all outputs of its matrix-vector multiplication. Every attainable matrix output is a linear combination of its columns.
Convex hull: The smallest convex set containing a given collection of points. It consists of convex combinations, not arbitrary linear combinations.
Vector addition: An operation that combines two vectors to produce another vector. The terms of a linear combination are joined by vector addition.
Coordinates: Numbers that specify an object's position relative to a chosen reference system or basis. Coefficients in a basis combination are the vector's coordinates.

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