k-frame

This article is about a linearly independent subset. For a spanning set in analysis, see frame of a vector space. For the structural component in motor vehicles, see Crossmember.

In linear algebra, a k-frame is an ordered set of k linearly independent vectors in a vector space; 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

• Frame (linear algebra)
• Frame of a vector space

Riemannian geometry

• Orthonormal frame
• Moving frame