Basis

Definition

A basis of $V$ is a list of vectors in $V$ that is linearly_independent and spans $V$.

Theorem: Criterion for basis

A list $v_{1},\ldots,v_{n}$ of vectors in $V$ is a basis of $V$ if and only if every $v \in V$ can be written uniquely in the form

$$ v = a_{1}v_{1} + \cdots + a_{n}v_{n} $$

where $a_{1},\ldots,a_{n} \in \mathbb{F}$.

Proof: See Axler, page 40.

Theorem: Spanning list contains a basis

Every spanning list in a vector_space can be reduced to a basis of the vector_space.

Theorem: Basis of finite-dimensional_vector_space

Every finite-dimensional_vector_space has a basis.

Theorem: linearly_independent list extends to a basis

Every linearly_independent list of vectors in a finite-dimensional_vector_space can be extended to a basis of the vector_space.