A vector space orientation is an equivalence class of ordered vector bases of a finite-dimensional real vector space 
. An ordered vector basis 
defines one such class. Another basis 
gives the same orientation if the matrix 
has a positive determinant,
in which case the basis 
is called oriented.
Every positive-dimensional finite-dimensional real vector space has two possible orientations
since the determinant of a nonsingular
matrix is either positive or negative.
For example, in 
,
represents one orientation and 
represents the other orientation. In
three dimensions, the cross
product uses the right-hand rule by convention,
reflecting the use of the canonical orientation 
as 
.
An orientation can be given by a nonzero element in the top exterior power of 
, i.e., 
. For example, 
gives the canonical orientation on 
and 
gives the other orientation.
Some special vector space structures imply an orientation. For example, if 
is a symplectic form
on 
,
of dimension 
, then 
gives an orientation. Also, if 
is a complex vector space,
then as a real vector space of dimension 
,
the complex structure gives an orientation.
