{{DISPLAYTITLE:''k''-frame}} {{about|a linearly independent subset|a spanning set in analysis|frame of a vector space|the structural component in motor vehicles|Crossmember}}{{One source|date=March 2026}} In linear algebra, a '''''k''-frame''' is an ordered set of ''k'' linearly independent vectors in a vector space<ref>{{Cite book |last=Tu |first=Loring W. |title=Introductory lectures on equivariant cohomology: with appendices by Loring W. Tu and Alberto Arabia |last2=Arabia |first2=Alberto |date=2020 |publisher=Princeton University Press |isbn=978-0-691-19748-7 |series=Annals of mathematics studies |location=Princeton |page=61}}</ref>; 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''<sub>''k''</sub>(''X''). * A ''k''-frame defines a parallelotope (a generalized parallelepiped); the volume can be computed via the Gram determinant.
== See also == * Frame (linear algebra) * Frame of a vector space
=== Riemannian geometry === * Orthonormal frame * Moving frame
== References == {{Reflist}}{{linear-algebra-stub}} Category:Linear algebra