Editing Orthonormal basis (section)

==As a principal homogeneous space==
{{Main|Stiefel manifold}}

The set of orthonormal bases for <math>\mathbb{R}^n</math> with the standard inner product is a [[principal homogeneous space]] or G-torsor for the [[orthogonal group]] <math>G = \text{O}(n),</math> and is called the [[Stiefel manifold]] <math>V_n(\R^n)</math> of orthonormal [[k-frame|<math>n</math>-frames]].<ref>{{Cite web|title=CU Faculty|url=https://engfac.cooper.edu/fred|access-date=2021-04-15|website=engfac.cooper.edu}}</ref>

In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point: given the space of orthonormal bases, there is no natural choice of orthonormal basis, but once one is given one, there is a one-to-one correspondence between bases and the orthogonal group.
Concretely, a linear map is determined by where it sends a given basis: just as an invertible map can take any basis to any other basis, an orthogonal map can take any ''orthogonal'' basis to any other ''orthogonal'' basis.

The other Stiefel manifolds <math>V_k(\R^n)</math> for <math>k < n</math> of ''incomplete'' orthonormal bases (orthonormal <math>k</math>-frames) are still homogeneous spaces for the orthogonal group, but not ''principal'' homogeneous spaces: any <math>k</math>-frame can be taken to any other <math>k</math>-frame by an orthogonal map, but this map is not uniquely determined.

* The set of orthonormal bases for <math>\mathbb{R}^{p,q}</math> is a G-torsor for <math>G = \text{O}(p,q)</math>.
* The set of orthonormal bases for <math>\mathbb{C}^n</math> is a G-torsor for <math>G = \text{U}(n)</math>.
* The set of orthonormal bases for <math>\mathbb{C}^{p,q}</math> is a G-torsor for <math>G = \text{U}(p,q)</math>.
* The set of right-handed orthonormal bases for <math>\mathbb{R}^n</math> is a G-torsor for <math>G = \text{SO}(n)</math>