Vector Space Orientation

An ordered vector basisv_1,...,v_n for a finite-dimensional vector space V defines an orientation. Another basis w_i=Av_i gives the same orientation if the matrix A has a positive determinant, in which case the basis w_i 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 R^2, {e_1,e_2} is one orientation and {e_2,e_1}∼{e_1,-e_2} is the other orientation. In three dimensions, the cross product uses the right-hand rule by convention, reflecting the use of the canonical orientation {e_1,e_2,e_3} as e_1×e_2=e_3.

An orientation can be given by a nonzero element in the top exterior power of V, i.e.,  ^ ^nV. For example, e_1 ^ e_2 ^ e_3 gives the canonical orientation on R^3 and -e_1 ^ e_2 ^ e_3 gives the other orientation.

Some special vector space structures imply an orientation. For example, if omega is a symplectic form on V, of dimension 2n, then omega^n gives an orientation. Also, if V is a complex vector space, then as a real vector space of dimension 2n, the complex structure gives an orientation.

See also

Bundle Orientation, Manifold Orientation

This entry contributed by Todd Rowland

Explore with Wolfram|Alpha

Cite this as:

Rowland, Todd. "Vector Space Orientation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

Subject classifications