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Basis

Definition

A subset $\mathcal{B} \subset V$ is a basis for $V$ if it is linearly independent and spanning.

Though we can construct bases for some infinite-dimensional vector spaces, we will assume in these notes that any basis is finite.

CAUTION! Remember that in linear algebra, we do not allow for infinite sums! This means that (for example) the set of singleton characteristic functions $\chi_\{x\}: \mathbb{R} \to \mathbb{R}$ is not a basis for the space of all functions $\mathbb{R} \to \mathbb{R}$.

Properties

The size of the basis is called the dimension of $V$, written $\dim V$.

Any spanning set for $V$ can be reduced to a basis. If $V$ has a basis, then any linearly independent set can be extended to a basis.

Related theorems

• The axiom of choice is equivalent to the statement that every vector space (including infinite ones) has a basis. (according to someone on stack exchange).

all these pages adapted with probably insufficient credit from my university's lecture notes