Let
V be a
finite-dimensional real vector space and let
b1 and
b2 be two ordered bases for
V. It is a standard result in
linear algebra that there exists a unique
linear transformation A :
V →
V that takes
b1 to
b2. The bases
b1 and
b2 are said to have the
same orientation (or be consistently oriented) if
A has positive
determinant; otherwise they have
opposite orientations. The property of having the same orientation defines an
equivalence relation on the set of all ordered bases for
V. If
V is non-zero, there are precisely two
equivalence classes determined by this relation. An
orientation on
V is an assignment of +1 to one equivalence class and −1 to the other. Every ordered basis lives in one equivalence class or another. Thus any choice of a privileged ordered basis for
V determines an orientation: the orientation class of the privileged basis is declared to be positive. For example, the
standard basis on
Rn provides a
standard orientation on
Rn (in turn, the orientation of the standard basis depends on the orientation of the
Cartesian coordinate system on which it is built). Any choice of a linear
isomorphism between
V and
Rn will then provide an orientation on
V. The ordering of elements in a basis is crucial. Two bases with a different ordering will differ by some
permutation. They will have the same/opposite orientations according to whether the
signature of this permutation is ±1. This is because the determinant of a
permutation matrix is equal to the signature of the associated permutation. Similarly, let
A be a nonsingular linear mapping of vector space
Rn to
Rn. This mapping is
orientation-preserving if its determinant is positive. For instance, in
R3 a rotation around the
Z Cartesian axis by an angle
α is orientation-preserving: \mathbf {A}_1 = \begin{pmatrix} \cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end{pmatrix} while a reflection by the
XY Cartesian plane is not orientation-preserving: \mathbf {A}_2 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{pmatrix}
Zero-dimensional case The concept of orientation degenerates in the zero-dimensional case. A zero-dimensional vector space has only a single point, the zero vector. Consequently, the only basis of a zero-dimensional vector space is the empty set \emptyset. Therefore, there is a single equivalence class of ordered bases, namely, the class \{\emptyset\} whose sole member is the
empty set. This means that an orientation of a
zero-dimensional space is a function \{\{\emptyset\}\} \to \{\pm 1\}. It is therefore possible to orient a point in two different ways, positive and negative. Because there is only a single ordered basis \emptyset, a zero-dimensional vector space is the same as a zero-dimensional vector space with ordered basis. Choosing \{\emptyset\} \mapsto +1 or \{\emptyset\} \mapsto -1 therefore chooses an orientation of every basis of every zero-dimensional vector space. If all zero-dimensional vector spaces are assigned this orientation, then, because all isomorphisms among zero-dimensional vector spaces preserve the ordered basis, they also preserve the orientation. This is unlike the case of higher-dimensional vector spaces where there is no way to choose an orientation so that it is preserved under all isomorphisms. However, there are situations where it is desirable to give different orientations to different points. For example, consider the
fundamental theorem of calculus as an instance of
Stokes' theorem. A closed interval is a one-dimensional
manifold with boundary, and its boundary is the set {{math|{
a,
b}}}. In order to get the correct statement of the fundamental theorem of calculus, the point should be oriented positively, while the point should be oriented negatively.
On a line The one-dimensional case deals with an
oriented line (or
directed line), which may be traversed in one of two directions. In
real coordinate space, an oriented line is also known as an
axis. There are two orientations to a
line (forward and backward) just as there are two orientations to an
oriented circle (clockwise and anti-clockwise). A semi-infinite oriented line is called a
ray (sometimes also a
half-axis or an
oriented half-line). In the case of a
line segment (a connected subset of a line), the two possible orientations result in
directed line segments.
On a surface An
orientable surface sometimes has the selected orientation indicated by the orientation of a
surface normal. An
oriented plane can be defined by a
pseudovector. ==Alternate viewpoints==