Editing Determinant (section)

=== Orientation of a basis ===
{{Main|Orientation (vector space)}}
The determinant can be thought of as assigning a number to every [[sequence]] of ''n'' vectors in '''R'''<sup>''n''</sup>, by using the square matrix whose columns are the given vectors. The determinant will be nonzero if and only if the sequence of vectors is a ''basis'' for '''R'''<sup>''n''</sup>. In that case, the sign of the determinant determines whether the [[orientation (vector space)|orientation]] of the basis is consistent with or opposite to the orientation of the [[standard basis]]. In the case of an orthogonal basis, the magnitude of the determinant is equal to the ''product'' of the lengths of the basis vectors.  For instance, an [[orthogonal matrix]] with entries in '''R'''<sup>''n''</sup> represents an [[orthonormal basis]] in [[Euclidean space]], and hence has determinant of ±1 (since all the vectors have length 1). The determinant is +1 if and only if the basis has the same orientation. It is −1 if and only if the basis has the opposite orientation.

More generally, if the determinant of ''A'' is positive, ''A'' represents an orientation-preserving [[linear transformation]] (if ''A'' is an orthogonal {{math|2 × 2}} or {{math|3 × 3}} matrix, this is a [[rotation (mathematics)|rotation]]), while if it is negative, ''A'' switches the orientation of the basis.