Editing Dimension (vector space) (section)

== Properties ==

If <math>W</math> is a [[linear subspace]] of <math>V</math> then <math>\dim (W) \leq \dim (V).</math>

To show that two finite-dimensional vector spaces are equal, the following criterion can be used: if <math>V</math> is a finite-dimensional vector space and <math>W</math> is a linear subspace of <math>V</math> with <math>\dim (W) = \dim (V),</math> then <math>W = V.</math> 

The space <math>\R^n</math> has the standard basis <math>\left\{e_1, \ldots, e_n\right\},</math> where <math>e_i</math> is the <math>i</math>-th column of the corresponding [[identity matrix]]. Therefore, <math>\R^n</math> has dimension <math>n.</math>

Any two finite dimensional vector spaces over <math>F</math> with the same dimension are [[isomorphic]]. Any [[bijective]] map between their bases can be uniquely extended to a bijective linear map between the vector spaces. If <math>B</math> is some set, a vector space with dimension <math>|B|</math> over <math>F</math> can be constructed as follows: take the set <math>F(B)</math> of all functions <math>f : B \to F</math> such that <math>f(b) = 0</math> for all but finitely many <math>b</math> in <math>B.</math> These functions can be added and multiplied with elements of <math>F</math> to obtain the desired <math>F</math>-vector space.

An important result about dimensions is given by the [[rank–nullity theorem]] for [[linear map]]s.

If <math>F / K</math> is a [[field extension]], then <math>F</math> is in particular a vector space over <math>K.</math> Furthermore, every <math>F</math>-vector space <math>V</math> is also a <math>K</math>-vector space. The dimensions are related by the formula
<math display=block>\dim_K(V) = \dim_K(F) \dim_F(V).</math>
In particular, every complex vector space of dimension <math>n</math> is a real vector space of dimension <math>2n.</math>

Some formulae relate the dimension of a vector space with the [[cardinality]] of the base field and the cardinality of the space itself.
If <math>V</math> is a vector space over a field <math>F</math> and if the dimension of <math>V</math> is denoted by <math>\dim V,</math> then:

:If dim <math>V</math> is finite then <math>|V| = |F|^{\dim V}.</math>
:If dim <math>V</math> is infinite then <math>|V| = \max (|F|, \dim V).</math>