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 $\mathrm{dim}(V)$, 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.

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$\beta =\{{e}_{i}\}$, $1\le i\le n$, is a basis for ${\mathbb{R}}^{n}$ (the $n$dimensional vector space over the reals). For $n=4$,
$$\beta =\{\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\right),\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\right),\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 1\hfill \\ \hfill 0\hfill \end{array}\right),\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 1\hfill \end{array}\right)\}$$ 
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$\beta =\{1,x,{x}^{2}\}$ is a basis for the vector space of polynomials with degree at most 2, over a division ring.

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The set
$$\beta =\{\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right],\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right],\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right],\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right]\}$$ 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
$${\beta}^{\prime}=\{\left[\begin{array}{cc}\hfill 2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right],\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right],\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 4\hfill \end{array}\right],\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill \frac{1}{2}\hfill & \hfill 0\hfill \end{array}\right]\}.$$ 
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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  20130322 12:01:57 
Last modified on  20130322 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 