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 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 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 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 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 an orthonormal basis

Given the central role under the Constitution of the Soviet Union, the party controlled all tiers of government and social institutions in the Soviet Union.

Given the extent and central location of his original holding from the Mill Seat Tract, and the fact that most of the town's pre-1944 building lots are to, Bare's ability to shape the size and extent of development in Roaring Spring was substantial.

Given the size of Merchant's capital outlay — including construction in Boswell of the world's largest tipple, a central business district and a high school — the company planned big things for Boswell.

Given his prestige, Rundstedt was a central figure in the Blomberg – Fritsch Affair which engulfed the German Army in early 1938.

Given its central location, Athlone is a natural hub for transport, with frequent bus and rail schedules.

Given that much of the first half of the novel focuses on the initial encounter, blossoming friendship and eventual estrangement of these central characters, this issue continues to pique the curiosity of readers.

Given no central organizational structure for Hinduism itself as a religion, various deviations from these norms do exist.

Given that G. atlantica is a small and G. stehlini a giant species, and that it is also not known when exactly the later evolved its large size, the ancestral lizards which colonized the central and western islands may have been still small or already large.

Given the city's central position in both Israeli nationalism ( Zionism ) and Palestinian nationalism, the selectivity required to summarise more than 5, 000 years of inhabited history is often influenced by ideological bias or background ( see Historiography and nationalism ).

Given central simple algebras A and B, one can look at the their tensor product A ⊗ B as a K-algebra ( see tensor product of R-algebras ).

Given two central simple algebras A ~ M ( n, S ) and B ~ M ( m, T ) over the same field F, A and B are called similar ( or Brauer equivalent ) if their division rings S and T are isomorphic.

Given the central role of TNF-alpha in many diseases, etanercept is being studied as treatment for a number of these disease, including over 150 clinical trials.

Given Zemlya i Volya's failures in its propaganda efforts among the peasants in the movements " to the people " in the early 1870s, Narodnaya Volya turned its energies against the central government.

Given such a choice of scale, a central representation of the Heisenberg group is a map of-algebras which is the formal way of saying that it sends the center to a chosen scale.

Given the central role the epic played over the centuries in Tibetan folk culture, Tibetan Buddhism has incorporated elements from it and interpreted them in religious terms.

Given Boston's natural and manmade geography, individuals from other neighborhoods in Boston must access the city's social center via one of the various bridges or tunnels that lead into central Boston.

# Given that rocky planets were ( at the time Rare Earth was written ) thought to form closer to their central stars, the planet probably forms too close to the star to lie within the habitable zone.

* Given, David R .-Vegetation on heated soils at Karapiti, central North Island, New Zealand, and its relation to ground temperature in New Zealand Journal of Botany, 1980, vol.

Given its central location, Ambato is an important industrial hub for Ecuador.

Given its status as a central bank, it is, however, governed by a number of special provisions, as laid down in the Nationalbank Act.

Given that VEGF plays a central role in the symptoms of POEMS syndrome, some have tried bevacizumab ( Avastin ), a monoclonal antibody directed against VEGF.

Together with Gary Breen, he formed an impressive central defensive partnership in front of Shay Given.

Barron noted: Given the central position he occupied in the intelligence-gathering network for the Mid-Ulster region, this RUC officer's interview with the Inquiry was disappointing.

Given the weakness of these pretexts, party chairman Gennadii Beliakov ascribed the rejection to the central government ’ s fears that the party would not be loyal to the president.