An ordered vector basis
for a finite-dimensional vector
space 
defines an orientation. Another basis 
gives the same orientation if the matrix 
has a positive determinant, in which case the basis 
is called oriented.
Any vector space has two possible orientations since the determinant of an nonsingular
matrix is either positive or negative. For example, in 
, 
is one orientation and 
is 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.
