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In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point: given the space of orthonormal bases, there is no natural choice of orthonormal basis, but once one is given one, there is a one-to-one correspondence between bases and the orthogonal group.
The Gram-Schmidt theorem, together with the axiom of choice, guarantees that every vector space admits an orthonormal basis. This is possibly the most significant use of orthonormality, as this fact permits operators on inner-product spaces to be discussed in terms of their action on the space's orthonormal basis vectors. What results is a deep ...
The concept of orthogonality may be extended to a vector space over any field of characteristic not 2 equipped with a quadratic form .Starting from the observation that, when the characteristic of the underlying field is not 2, the associated symmetric bilinear form , = ((+) ()) allows vectors and to be defined as being orthogonal with respect to when (+) () = .
In finite-dimensional spaces, the matrix representation (with respect to an orthonormal basis) of an orthogonal transformation is an orthogonal matrix. Its rows are mutually orthogonal vectors with unit norm, so that the rows constitute an orthonormal basis of V. The columns of the matrix form another orthonormal basis of V.
With respect to the standard basis e 1, e 2, e 3 of the columns of R are given by (Re 1, Re 2, Re 3). Since the standard basis is orthonormal, and since R preserves angles and length, the columns of R form another orthonormal basis. This orthonormality condition can be expressed in the form
A conformal map acting on a rectangular grid. Note that the orthogonality of the curved grid is retained. While vector operations and physical laws are normally easiest to derive in Cartesian coordinates, non-Cartesian orthogonal coordinates are often used instead for the solution of various problems, especially boundary value problems, such as those arising in field theories of quantum ...
An orthonormal basis is a basis whose vectors are both orthogonal and normalized (they are unit vectors). A conformal linear transformation preserves angles and distance ratios, meaning that transforming orthogonal vectors by the same conformal linear transformation will keep those vectors orthogonal .
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q T Q = Q Q T = I , {\displaystyle Q^{\mathrm {T} }Q=QQ^{\mathrm {T} }=I,} where Q T is the transpose of Q and I is the identity matrix .