Basis (linear algebra)
A basis is a linearly independent spanning set of a vector space.
In mathematics, a basis (pl.: bases) of a vector space V is a set B of elements such that every element of V can be written uniquely as a finite linear combination of elements of B. The coefficients of this combination are called coordinates or components of the vector with respect to B. Equivalently, a basis is a linearly independent spanning set. A vector space can have several bases, but all bases have the same number of elements, called the dimension of the vector space. This concept is fundamental to linear algebra and applies to both finite-dimensional and infinite-dimensional vector spaces.
- Field
- Mathematics (linear algebra)
- Known for
- Definition of basis, linear independence, spanning set, coordinates, dimension of vector spaces
Lore & Background
A basis B of a vector space V over a field F (such as the real numbers or complex numbers) is defined by two properties: linear independence and spanning. Linear independence means that for any finite subset of B, the only linear combination equal to the zero vector has all scalar coefficients zero. The spanning property means every vector in V can be written as a linear combination of some vectors in B. Together, these ensure that each vector has a unique representation in terms of the basis vectors.
Reader's Guide
The concept of a basis is central to linear algebra because it provides a coordinate system for vector spaces. For finite-dimensional spaces, the number of basis vectors is the dimension, a key invariant. Bases are not unique; for example, in R², both the standard basis {(1,0), (0,1)} and the set {(1,1), (-1,2)} are valid bases. When an ordering is assigned to basis vectors, one speaks of an ordered basis, which is necessary for associating coordinates with specific basis elements. Basis vectors find applications in the study of crystal structures and frames of reference. The principles of bases extend to infinite-dimensional spaces, though the article focuses mainly on finite-dimensional cases.
Did You Know?
- The coefficients of the unique linear combination of basis vectors representing a vector are called its coordinates.
- In R², the vectors (1,0) and (0,1) form the standard basis, but any pair of linearly independent vectors also forms a basis.
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