Editing Operator (mathematics) (section)

=== Geometry ===
{{Main|General linear group|Isometry}}

In [[geometry]], additional structures on [[vector space]]s are sometimes studied. Operators that map such vector spaces to themselves [[bijective]]ly are very useful in these studies, they naturally form [[group (mathematics)|group]]s by composition.

For example, bijective operators preserving the structure of a vector space are precisely the [[invertible function|invertible]] [[linear operator]]s. They form the [[general linear group]] under composition. However, they ''do not'' form a vector space under operator addition; since, for example, both the identity and −identity are [[invertible]] (bijective), but their sum, 0, is not.

Operators preserving the [[Euclidean metric]] on such a space form the [[isometry group]], and those that fix the origin form a subgroup known as the [[orthogonal group]]. Operators in the orthogonal group that also preserve the orientation of vector tuples form the [[special orthogonal group]], or the group of rotations.