Frame (linear algebra)

In mathematics, a frame of a vector space V, is either of two distinct notions, both generalizing the notion of a basis:

• In one definition, a k-frame is an ordered set of k linearly independent vectors in a space; thus k ≤ n the dimension of the vector space, and if k = n 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.

• In the other definition, a frame is a certain type of ordered set of vectors that spans a space. Thus k ≥ n.

These are rarely confused and generally clear from context, as the former is a basic concept in finite-dimensional geometry, such as Stiefel manifolds, while the latter is most used in analysis. Further, the former must have at most as many elements as the dimension of the space, while the latter must have at least as many elements as the dimension of the space, so the only overlapping sets are bases.

See also

• k-frame
• Frame of a vector space

Riemannian geometry

• Orthonormal frame
• Moving frame
• Overcompleteness