Linear algebra MOC

Basis of a vector space

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

Basis proofs

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

  1. 𝐡 βŠ†π‘‰
  2. 𝐡 is linearly dependent
  3. 𝑉 =span⁑(𝐡)

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 ↩