Let "V" be a finite-dimensional real vector space and let "b"1 and "b"2 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 "b"1 to "b"2. The bases "b"1 and "b"2 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.