\[ \newcommand{\d}{\mathrm{d}} \newcommand{\e}{\mathrm{e}} \newcommand{\i}{\mathrm{i}} \]

Given a finite dimensional vector space \(V\). We call two ordered basis \(B\) and \(B'\) equivalent if the transformation matrix has positive determinant .

One can prove, that only two equivalence classes exists. We choose one and call all ordered basis of this particular equivalence class positive oriented and basis from the other equivalence class negative oriented.

A vector space equipped with such an equivalence class is called oriented vector space.

See also Link to heading

• orientation on manifolds