[[Linear algebra MOC]]
# Basis of a vector space
Given a [[Vector space]] $V$,
a (Hammel) **basis** $\mathcal{B}$ is a [[Linear (in)dependence|linearly independent]] [[Span|spanning set]] of $V$.[^2008] #m/def/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.

> [!tip]- Basis proofs
> To prove that a set $B$ is a basis for the space $V$, it is necessary to show the following:
> 1. $B \sube V$
> 2. $B$ is linearly dependent
> 3. $V = \Span(B)$

[^2008]: 2008\. [[Sources/@romanAdvancedLinearAlgebra2008|Advanced Linear Algebra]], p. 47

See also [[Dense basis]].

## Properties

1. [[Every vector space has a basis]]

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