PlanetMath: basis

(more info)

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basis

(Definition)

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 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.

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

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

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

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

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