Frame (linear algebra)
Encyclopedia
In mathematics
, a frame of a vector space
V, is either of two distinct notions, both generalizing the notion of a basis
:
These are rarely confused and generally clear from context, as the former is a basic concept in finite-dimensional geometry, such as Stiefel manifold
s, 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a frame of a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
V, is either of two distinct notions, both generalizing the notion of a basis
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
:
- In one definition, a k-frameK-frameIn linear algebra, a branch of mathematics, a k-frame is an ordered set of k linearly independent vectors in a space; thus k ≤ n, where n is 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,...
is an ordered setOrdered setIn order theory in mathematics, a set with a binary relation R on its elements that is reflexive , antisymmetric and transitive is described as a partially ordered set or poset...
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 frameOrthonormal frameIn Riemannian geometry and relativity theory, an orthonormal frame is a tool for studying the structure of a differentiable manifold equipped with a metric...
, respectively.
- If the vectors are orthogonal or orthonormal, the frame is called an orthogonal frame or orthonormal frame
- In the other definition, a frameFrame of a vector spaceIn linear algebra, a frame of a vector space V with an inner product can be seen as a generalization of the idea of a basis to sets which may be linearly dependent...
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 manifold
Stiefel manifold
In mathematics, the Stiefel manifold Vk is the set of all orthonormal k-frames in Rn. That is, it is the set of ordered k-tuples of orthonormal vectors in Rn. It is named after Swiss mathematician Eduard Stiefel...
s, 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.