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Given a pre-Hilbert space , an orthonormal basis for is an orthonormal set of vectors with the property that every vector in can be written as an infinite linear combination of the vectors in the basis. In this case, the orthonormal basis is sometimes called a Hilbert basis for . Note that an orthonormal basis in this sense is not generally a ...
If an orthonormal basis is to be produced, then the algorithm should test for zero vectors in the output and discard them because no multiple of a zero vector can have a length of 1. The number of vectors output by the algorithm will then be the dimension of the space spanned by the original inputs.
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 ...
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 .
The solution can then be expressed as ^ = (), where is an matrix containing the first columns of the full orthonormal basis and where is as before. Equivalent to the underdetermined case, back substitution can be used to quickly and accurately find this x ^ {\displaystyle {\hat {\mathbf {x} }}} without explicitly inverting R 1 {\displaystyle R ...
The geometric content of the SVD theorem can thus be summarized as follows: for every linear map : one can find orthonormal bases of and such that maps the -th basis vector of to a non-negative multiple of the -th basis vector of , and sends the leftover basis vectors to zero.
The space ℓ ∞ is not separable, and therefore has no Schauder basis. Every orthonormal basis in a separable Hilbert space is a Schauder basis. Every countable orthonormal basis is equivalent to the standard unit vector basis in ℓ 2. The Haar system is an example of a basis for L p ([0, 1]), when 1 ≤ p < ∞. [2]
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.