Given a pre-Hilbert space H, an orthonormal basis for H is an orthonormal set of vectors with the property that every vector in H can be written as an infinite linear combination of the vectors in the basis.

Given that the eigenstates of the Hamiltonian always from a set of orthonormal basis, both the cases of m

Given an orthonormal basis ordered such that, we see that

Given a C < sup > n + 1 </ sup >- curve γ in R < sup > n </ sup > which is regular of order n the Frenet frame for the curve is the set of orthonormal vectors

Given an orthonormal basis, any pure state of a two-level quantum system can be written as a superposition of the basis vectors

Given the frame field, one can also define a metric by conceiving of the frame field as an orthonormal vector field.

If V is finite-dimensional, then V * has the same dimension as V. Given a basis

Given that estimation is undertaken on the basis of a least squares analysis, estimates of the unknown parameters β < sub > j </ sub > are determined by minimising a sum of squares function

Given a basis of a vector space, every element of the vector space can be expressed uniquely as a finite linear combination of basis vectors.

Given the complexity of NLP problems, it is often difficult to predict performance only on the basis of glass-box evaluation, but this type of evaluation is more informative with respect to error analysis or future developments of a system.

Given Poseidon's connection with horses as well as the sea, and the landlocked situation of the likely Indo-European homeland, Nobuo Komita has proposed that Poseidon was originally an aristocratic Indo-European horse-god who was then assimilated to Near Eastern aquatic deities when the basis of the Greek livelihood shifted from the land to the sea, or a god of fresh waters who was assigned a secondary role as god of the sea, where he overwhelmed the original Aegean sea deities such as Proteus and Nereus.

Given the medieval medicine's main theory of humorism, spices and herbs were indispensable to balance " humors " in food, a daily basis for good health at a time of recurrent pandemics.

The isomorphism is not canonical ; it depends on a choice of basis in V. Given a basis ( e < sub > 1 </ sub >, ..., e < sub > n </ sub >) of V and an automorphism T in GL ( V ), we have

Given that there is no constitutional office of Deputy Prime Minister, with the position being recreated on a case by case basis, the person who holds the post has no official residence.

Given a basis f = ( X < sub > 1 </ sub >,..., X < sub > n </ sub >) of V, there is a unique reciprocal basis f < sup >#</ sup > = ( Y < sup > 1 </ sup >,..., Y < sup > n </ sup >) of V determined by requiring that

Given the Protestant belief of justification through faith alone, versus on the basis of merit, most Protestant Christians consider themselves saved without having to keep the commandments of the Mosaic Law as a whole.

Given that it is still legal to discriminate against individuals on the basis of gender identity within New Zealand, the transgender community often finds that many of its younger members require survival sex for food, shelter and rest.

Given a set S of matrices, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of S. Equivalently, for any set S of mutually commuting semisimple linear transformations of a finite-dimensional vector space V there exists a basis of V consisting of simultaneous eigenvectors of all elements of S. Each of these common eigenvectors v ∈ V, defines a linear functional on the subalgebra U of End ( V ) generated by the set of endomorphisms S ; this functional is defined as the map which associates to each element of U its eigenvalue on the eigenvector v. This " generalized eigenvalue " is a prototype for the notion of a weight.

Given a set, a free module on is a free module with basis.

For a finite group G, the left regular representation λ ( over a field K ) is a linear representation on the K-vector space V whose basis is the elements of G. Given g ∈ G, λ ( g ) is the linear map determined by its action on the basis by left translation by g, i. e.

Given the proximity of Boarstall to the king's palace at Brill it would appear that this legend certainly has some basis in fact.

Given the Likert Scale's ordinal basis, summarizing the central tendency of responses from a Likert scale by using either the median or the mode is best, with ‘ spread ’ measured by quartiles or percentiles.

** The theorem that every Hilbert space has an orthonormal basis.

For a finite-dimensional vector space, using a fixed orthonormal basis, the inner product can be written as a matrix multiplication of a row vector with a column vector:

One orthonormal basis is:

Consider a complete orthonormal system ( basis ),, for a Hilbert space H, with respect to the norm from an inner product.

The transform is sometimes denoted by the symbol, as in or or < ref group =" note "> As a linear transformation on a finite-dimensional vector space, the DFT expression can also be written in terms of a DFT matrix ; when scaled appropriately it becomes a unitary matrix and the X < sub > k </ sub > can thus be viewed as coefficients of x in an orthonormal basis .</ ref >

Hilbert spaces can be completely classified: there is a unique Hilbert space up to isomorphism for every cardinality of the orthonormal basis.

In particular, Banach spaces lack a notion analogous to an orthonormal basis.

Applying Gram-Schmidt one obtains an orthonormal basis for.

By transferring upper triangularisation of operators of finite-dimensional complex vector space, there is an internal orthonormal Hilbert space basis for where runs from to, such that each of the corresponding-dimensional subspaces is-invariant.

* A Hilbert space is separable if and only if it has a countable orthonormal basis, it follows that any separable, infinite-dimensional Hilbert space is isometric to ℓ < sup > 2 </ sup >.

This is an algebra built up from an orthonormal basis of mutually orthogonal vectors under addition and multiplication, p of which have norm + 1 and q of which have norm − 1, with the product rule for the basis vectors

Choose an orthonormal basis for V i. e. where and for.

In this construction the representation of the Clifford algebra Cℓ ( V, g ), the Lie algebra so ( V, g ), and the Spin group Spin ( V, g ), all depend on the choice of the orthonormal basis and the choice of the gamma matrices.

form an orthonormal basis.

When viewed in this way the numbers x < sub > 1 </ sub >, ..., x < sub > n </ sub >, and y < sub > 1 </ sub >, ..., y < sub > n </ sub > are the components of x and y with respect to an orthonormal basis of V.

More precisely, recoordinatize R < sup > n </ sup > with any orthonormal basis whose first two vectors span a subspace containing x and y.

Let P < sup >− 1 </ sup > DP be an eigendecomposition of M, where P is a unitary complex matrix whose rows comprise an orthonormal basis of eigenvectors of M, and D is a real diagonal matrix whose main diagonal contains the corresponding eigenvalues.

In formal terms, this representation is a wavelet series representation of a square-integrable function with respect to either a complete, orthonormal set of basis functions, or an overcomplete set or frame of a vector space, for the Hilbert space of square integrable functions.

* In functional analysis, a subset S of a topological vector space V is complete if its span is dense in V. In the particular case of Hilbert spaces ( or more generally, inner product spaces ), an orthonormal basis is a set that is both complete and orthonormal.

Then the columns of V such that the corresponding form an orthonormal basis of the nullspace of A.

Let the curve be a unit speed curve and let t = u × T so that T, u, t form an orthonormal basis: the Darboux frame.

This basis is not orthonormal and it is difficult to use, but now the velocity vector can be changed to the fixed frame or to the moving frame with just a change of bases.

Technically they form an orthonormal eigenvector basis, satisfying