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.

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

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

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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

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

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

$\beta^{\prime}=\left\{\begin{bmatrix}2&0\\ 0&0\end{bmatrix},\begin{bmatrix}0&1\\ 0&0\end{bmatrix},\begin{bmatrix}0&0\\ 0&4\end{bmatrix},\begin{bmatrix}0&0\\ \frac{1}{2}&0\end{bmatrix}\right\}.$

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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	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