Linear combination

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In a vector space, a linear combination of vectors is a sum of scalar multiples of the vectors. Every expression created by combining vectors using the addition and scalar multiplication operations can be simplified to a linear combination of distinct vectors. Linear combinations are for this reason often used as a stand-in whenever one expressions and equations in a vector space.

Many concepts in the theory of vector spaces are most easily expressed through linear combinations. For instance, a basis of a vector space can be defined as a set of vectors in the space with the property that every vector can be uniquely expressed as a linear combination of the basis vectors. A linear transformation can be defined briefly as a function between vector spaces that "preserves linear combinations".