k-frame

In linear algebra, a k-frame is an ordered set of k linearly independent vectors in a vector space^[1]; thus, k ≤ n, where n is the dimension of the space, and an n-frame is precisely an ordered basis.

If the vectors are orthogonal, or orthonormal, the frame is called an orthogonal frame, or orthonormal frame, respectively.

Properties

The set of k-frames (particularly the set of orthonormal k-frames) in a given vector space X is known as the Stiefel manifold, and denoted V_k(X).
A k-frame defines a parallelotope (a generalized parallelepiped); the volume can be computed via the Gram determinant.

See also

Riemannian geometry

References

^ Tu, Loring W.; Arabia, Alberto (2020). Introductory lectures on equivariant cohomology: with appendices by Loring W. Tu and Alberto Arabia. Annals of mathematics studies. Princeton: Princeton University Press. p. 61. ISBN 978-0-691-19748-7.