Dimension (vector space)

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the dimension of a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> V is the <a href="/facts/Cardinality/m6VPojwT">cardinality</a> (i.e., the number of vectors) of a <a href="/facts/Basis_(linear_algebra)/89IPoN6c">basis</a> of V over its base <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a>. It is sometimes called Hamel dimension (after <a href="/facts/Georg_Hamel/jWMPpcrw">Georg Hamel</a>) or algebraic dimension to distinguish it from other types of <a href="/facts/Dimension/0ktmUyac">dimension</a>.
For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined. We say 
 
 
 
 V
 
 
 {\displaystyle V}
 
 is finite-dimensional if the dimension of 
 
 
 
 V
 
 
 {\displaystyle V}
 
 is finite, and infinite-dimensional if its dimension is <a href="/facts/Infinity/cF506uD7">infinite</a>.
The dimension of the vector space 
 
 
 
 V
 
 
 {\displaystyle V}
 
 over the field 
 
 
 
 F
 
 
 {\displaystyle F}
 
 can be written as 
 
 
 
 
 dim
 
 F
 
 
 ⁡
 (
 V
 )
 
 
 {\displaystyle \dim _{F}(V)}
 
 or as 
 
 
 
 [
 V
 :
 F
 ]
 ,
 
 
 {\displaystyle [V:F],}
 
 read "dimension of 
 
 
 
 V
 
 
 {\displaystyle V}
 
 over 
 
 
 
 F
 
 
 {\displaystyle F}
 
". When 
 
 
 
 F
 
 
 {\displaystyle F}
 
 can be inferred from context, 
 
 
 
 dim
 ⁡
 (
 V
 )
 
 
 {\displaystyle \dim(V)}
 
 is typically written.

Dimension (vector space) open-in-new

Dimension (vector space)