Linear algebra MOC

Basis of a vector space

Given a Vector space over , a (Hammel) basis is a linearly independent spanning set of ,1 #m/def/linalg or equivalently, an linear isomorphism for some free vector space . A basis is particularly useful in the form of an Ordered basis, allowing for vectors and linear maps to be represented as coördinate matrices.

Basis proofs

To prove that a set is a basis for the space , it is necessary to show the following:

  2. is linearly dependent

See also Dense basis.

Properties

  1. Assuming choice, every vector space has a basis

#state/tidy | #SemBr | #lang/en

Footnotes

  1. 2008. Advanced Linear Algebra, p. 47 ↩