1.2.11 Linear Combination Definition
A linear combination is a fundamental concept in algebra, combining vectors or variables using scalar coefficients to form new mathematical expressions.
Linear Combination Definition is the characterization of a linear combination as an expression formed by taking a finite collection of vectors from a vector space, multiplying each one by a scalar coefficient drawn from the underlying field, and summing the results, producing a new vector that depends on both the chosen vectors and the chosen coefficients. It supplies the single algebraic operation from which the notions of spanning, linear independence, and basis are all constructed, making it one of the earliest and most heavily used concepts in the foundations of tensor algebra.
The Basic Construction
Given any finite list of vectors from a vector space and a corresponding list of scalars from the underlying field, a linear combination is formed by multiplying each vector by its associated scalar and adding all of the resulting vectors together. Because vector addition and scalar multiplication are both operations guaranteed to exist by the vector space axioms, this construction is always well defined, regardless of which vectors and scalars are chosen, and it always produces another vector belonging to the same space.
The expression above defines a linear combination of k vectors, where each coefficient is a scalar drawn from the field over which the vector space is defined.
Special Cases
Two linear combinations play a distinguished role in the surrounding theory. The trivial linear combination, in which every coefficient is chosen to equal zero, always produces the zero vector, regardless of which vectors are used; this trivial case is central to the definition of linear independence, which requires that the trivial linear combination be the only one that produces the zero vector among a given set of vectors. A nontrivial linear combination, by contrast, is one in which at least one coefficient is nonzero, and the existence of a nontrivial linear combination equaling the zero vector among a set of vectors is precisely what it means for that set to be linearly dependent.
Linear Combinations, Spanning, and Bases
The set of every possible linear combination of a given collection of vectors is called the span of that collection, and it always forms a subspace of the original vector space. A collection of vectors is said to span the whole space if every vector in the space can be obtained as some linear combination of the collection. Combined with linear independence, spanning is one of the two conditions required for a collection of vectors to qualify as a basis, and it is precisely because every vector in the space can be written as a linear combination of a basis that the coordinates of a vector relative to that basis are well defined.
Linear Combinations Beyond Vectors
The concept of a linear combination is not confined to vectors alone; it applies equally to any objects belonging to a vector space, including covectors in a dual space, matrices, polynomials, functions satisfying an appropriate vector space structure, and tensors of any rank, since the collection of all tensors of a given type itself forms a vector space under tensor addition and scalar multiplication. In every one of these settings, the definition of a linear combination remains the same: a sum of scaled elements, with the scaling drawn from the relevant scalar field.
Why Linear Combinations Matter for Tensor Algebra
Linear combinations are the mechanism through which every vector space, and every space built from vector spaces — including dual spaces, tensor product spaces, and the tensor algebra itself — is generated and organized. Whenever a vector, covector, or tensor is expressed relative to a chosen basis, that expression is a linear combination, and the transformation rules that describe how such expressions change when the basis changes are, at their core, statements about how the coefficients of a linear combination must be recalculated to represent the same underlying object relative to a different set of basis elements. Understanding linear combinations precisely is therefore a prerequisite for understanding coordinates, bases, and ultimately the tensor transformation law itself.