Basis (linear algebra)

A basis is a linearly independent set of vectors that spans a vector space. Every vector in the space has a unique expression as a linear combination of basis vectors.

Linear independence: A set of vectors is linearly independent when only the trivial linear combination equals zero. Independence prevents basis vectors from being redundant.
Coordinates: Coordinates are the scalars that specify a vector relative to an ordered basis. Each vector's unique basis expansion becomes a coordinate tuple.
Generating set: A generating set is a collection of vectors whose linear combinations fill a specified vector space. A generating set may span the space while containing redundant vectors.
Standard basis: The standard basis of coordinate space consists of vectors with one entry equal to one and all others zero. It is the familiar reference basis underlying ordinary coordinate notation.
Span (linear algebra): The span of a set of vectors is the set of all their linear combinations. A basis must span the entire vector space.
Change of basis: A change of basis converts coordinate descriptions of the same vectors between two bases. It translates representations without changing the underlying vectors.
Orthonormal basis: An orthonormal basis is a basis whose vectors are mutually orthogonal and have unit length. It adds geometric structure that makes coordinates and projections simpler.
Eigenbasis: An eigenbasis is a basis consisting entirely of eigenvectors of a linear operator. In this basis, a diagonalizable operator acts by scaling each coordinate separately.
Vector space: A vector space is a set with addition and scalar multiplication satisfying specified algebraic laws. A basis is defined relative to the space it spans.
Coordinate vector: A coordinate vector is the column of coefficients expressing a vector in an ordered basis. It packages a basis expansion into a standard algebraic object.

Linked from 33 pages