Basis (linear algebra): definition, properties, examples, and uses

Basic properties and consequences

When a vector space has a basis, every element of the space has a unique representation as a linear combination of basis vectors. The number of vectors in any basis (if finite) is the dimension of the space. Key equivalent characterizations include: a basis is a linearly independent spanning set, a minimal spanning set, or a maximal linearly independent set. For more on the surrounding concepts see vector space and the definitions of span and linear independence.

Examples

• The standard basis of R^n consists of n vectors each having a single 1 and zeros elsewhere; it both spans R^n and is independent.
• Polynomials of degree less than n have the monomials 1, x, x^2, ..., x^{n-1} as a basis for that finite-dimensional space.
• Function spaces and infinite-dimensional spaces may have bases with infinitely many elements; such examples illustrate how bases generalize beyond simple Euclidean spaces.

History and existence

The concept of a basis formalizes the idea of coordinates and was developed as linear algebra matured in the 19th century. In general, every vector space has a basis, but proving existence for arbitrary (possibly infinite-dimensional) spaces relies on set-theoretic principles such as Zorn's lemma or the axiom of choice. The size (cardinality) of any two bases of the same space is always the same; this invariant is the dimension, which can be finite or infinite.

Uses and related constructions

Bases enable coordinate representations, which reduce abstract vector problems to numerical calculations. Changing from one basis to another is managed by a change-of-basis matrix; orthonormal bases simplify computations in inner product spaces and are central in numerical methods and signal processing. Standard computational tasks—solving linear systems, diagonalizing operators, computing projections—frequently begin by choosing a convenient basis. See coordinate systems and change of basis for practical techniques.

Notable distinctions and facts

While all bases share the core definition, there are important variations: orthonormal bases (in inner product spaces), Hamel bases (for algebraic spanning in infinite-dimensional spaces), and Schauder bases (in topological vector spaces where convergence matters). For foundational reading on dimension and bases consult dimension and introductory linear algebra texts.