# basis

A (Hamel) basis of a vector space is a linearly independent spanning set.

It can be proved that any two bases of the same vector space must have the same cardinality. This introduces the notion of dimension of a vector space, which is precisely the cardinality of the basis, and is denoted by $\operatorname{dim}(V)$, where $V$ is the vector space.

The fact that every vector space has a Hamel basis (http://planetmath.org/EveryVectorSpaceHasABasis) is an important consequence of the axiom of choice (in fact, that proposition is equivalent to the axiom of choice.)

Examples.

• $\beta=\{e_{i}\}$, $1\leq i\leq n$, is a basis for $\mathbb{R}^{n}$ (the $n$-dimensional vector space over the reals). For $n=4$,

 $\beta=\left\{\begin{pmatrix}1\\ 0\\ 0\\ 0\end{pmatrix},\begin{pmatrix}0\\ 1\\ 0\\ 0\end{pmatrix},\begin{pmatrix}0\\ 0\\ 1\\ 0\end{pmatrix},\begin{pmatrix}0\\ 0\\ 0\\ 1\end{pmatrix}\right\}$
• $\beta=\{1,x,x^{2}\}$ is a basis for the vector space of polynomials with degree at most 2, over a division ring.

• The set

 $\beta=\left\{\begin{bmatrix}1&0\\ 0&0\end{bmatrix},\begin{bmatrix}0&1\\ 0&0\end{bmatrix},\begin{bmatrix}0&0\\ 0&1\end{bmatrix},\begin{bmatrix}0&0\\ 1&0\end{bmatrix}\right\}$

is a basis for the vector space of $2\times 2$ matrices over a division ring, and assuming that the characteristic of the ring is not 2, then so is

 $\beta^{\prime}=\left\{\begin{bmatrix}2&0\\ 0&0\end{bmatrix},\begin{bmatrix}0&1\\ 0&0\end{bmatrix},\begin{bmatrix}0&0\\ 0&4\end{bmatrix},\begin{bmatrix}0&0\\ \frac{1}{2}&0\end{bmatrix}\right\}.$
• The empty set is a basis for the trivial vector space which consists of the unique element $0$.

Remark. More generally, for any (left) right module $M$ over a ring $R$, one may define a (left) right basis for $M$ as a subset $B$ of $M$ such that $B$ spans $M$ and is linearly independent. However, unlike bases for a vector space, bases for a module may not have the same cardinality.

 Title basis Canonical name Basis Date of creation 2013-03-22 12:01:57 Last modified on 2013-03-22 12:01:57 Owner mathcam (2727) Last modified by mathcam (2727) Numerical id 22 Author mathcam (2727) Entry type Definition Classification msc 15A03 Synonym Hamel basis Related topic Span Related topic IntegralBasis Related topic BasicTensor Related topic Aliasing Related topic Subbasis Related topic Blade Related topic ProofOfGramSchmidtOrthogonalizationProcedure Related topic LinearExtension