You are here
Homebasis
Primary tabs
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 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.
Mathematics Subject Classification
15A03 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections