Linear algebra MOC

Basis of a vector space

Given a Vector space 𝑉, a (Hammel) basis B is a linearly independent spanning set of 𝑉.¹ linalg 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 𝐵 is a basis for the space 𝑉, it is necessary to show the following:

1. 𝐵 ⊆𝑉
2. 𝐵 is linearly dependent
3. 𝑉 =span⁡(𝐵)

See also Dense basis.

Properties

1. Every vector space has a basis

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Footnotes

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