Dual vector space

Exercise 2.14 (Used in Section 5.5) Let V denote a complex vector space. Let y denote the set of complex linear transformations from V to C. Show that y is a complex vector space. Show that ifV is finite dimensional then dim y = dim V. The vector space y is called the dual vector space or, more simply, the dual space. 

Exercise 3.19 (Used in Exercises 5.21 and 5.22) Suppose V is a finite-dimensional complex scalar product space. Recall the dual vector space V from Exercise 2.14. Consider the function t V V defined by... 

To define the dual representation we first must define dual vector spaces. 

Definition 5.5 Suppose V is a complex vector space. The dual vector space of V, denoted V and pronounced "V -dual, is the complex vector space of linear transformations from V to C. 

Remark 10. 93s is formed with respect to a given set of species. and is not to be confused with the dual vector-space to 9lT. 

Proof. Consider the dual vector space 93s- This is isomorphic to an S dimensional vector space, 5ls, in which 2IT is embedded. Let b be the subspace of 93s which is isomorphic to the subspace of Us within which the set s/s lies. By Theorem 2 the dimension of b is t. Now the proper reactions of 93s are the subspace b of annihilators of b. By a well-known theorem on dual vector spaces [d], b must have dimension (s - t ). ... 

Then the vector space Tx,x and the corresponding scheme will both be called the tangent space to X at x. The dual vector space mx/mx, and its scheme (mx/ml)Bch are both the cotangent space to X at x. 

Postulate B.—There exists a set of vectors indicated by the bra symbol < in one-to-one correspondence with the vectors of 3/f, forming a dual Hilbert space 3 . As a matter of notation we use the symbol , etc. This dual space must be such that a meaning can be given to the scalar product of any vector with the following properties ... 

In order to be consistent with the concept of the two dual data spaces, we must also define the vector of row-means... 

Since U and V express one and the same set of latent vectors, one can superimpose the score plot and the loading plot into a single display as shown in Fig. 31,2e. Such a display was called a biplot (Section 17.4), as it represents two entities (rows and columns of X) into a single plot [10]. The biplot plays an important role in the graphic display of the results of PCA. A fundamental property of PCA is that it obviates the need for two dual data spaces and that instead of these it produces a single space of latent variables. 

In terms of this notation the physical state of a system is represented in a dual complex vector space by kets, like a) and bras, (a. The state ket (or bra) carries complete information about the physical state. The sum of two kets is another ket. 

The concept of quantum states is the basic element of quantum mechanics the set of quantum states I > and the field of complex numbers, C, define a Hilbert space as being a linear vector space the mapping < P P> introduces the dual conjugate space (bra-space) to the ket-space the number C(T )=< I P> is a... 

Exercise 2.15 (Used in Proposition 11.1) Suppose V is a finite-dimensional complex vector space. Show that V = (y ), (See Exercise 2.14 for a definition of the dual V. ) Is this true for all complex vector spaces ... 

In Section 4.4 we saw how to build a representation from the action of a group on a set the new representation space is a space of functions. In this section, we apply this idea to linear functions on a vector space of a representation to define the dual representation. 

So when there is one fixed complex scalar product on a vector space V, it is consistent to use the notation v for both dual and adjoint. In a unitary basis, the asterisk means coordinate transpose. 

In this section we have shown how a representation on a vector space determines a representation on the dual of the vector space. We will find the dual representation useful in Section 5,5, More generally, duality is an important theoretical concept in many mathematical settings. Physically, momentum space is dual to position space, so the name "momentum space in the physics literature often connotes duality. 

The set of bra functionals form in themselves a vector space similar to S, and called the dual to S. The elements are also called bra vectors. In order to get a simple and consistent notation, we also change the notation of the usual vectors Where we used to write simply x, we now write x), which is called a ket vector. The names are due to Dirac and refen to the common expression (x Cy), now written as [Pg.5]

A dual space or adjoint space of a vector space X, denoted X, is the space of all functions on X. 

Compared with the Dirac s bra-ket formalism, diagonal vector spaces, except for the possibility to have their elements conjugated, lack of the dual space distinction as bra-ket or row-colunm vectors have. A possible way to overcome this situation is to construct bra-ket diagonal matrices with the aid of a set of auxiliary matrices adopting the appropriate dimension. The following definition will be useful. 

At the tip contact, the motion and constraint vector spaces may be defined using the general joint model discussed in Section 2.3. For convenience, we will assume that the two dual bases used to partitiai the spatial acceleration and force vectors at the tip. 

As regards the importance vector, the theory follows quite directly by a passage from the space of expected neutron distributions N to the dual ( adjoint ) space A of linear functionals. The effect of P and Q on this dual space is expressed by their transposes P and Q, respectively, as always. 

Theorem 4.2.6 motivates the study on Lie algebras of pairs of Poisson brackets which appear from two different structures of Lie algebras on one vector space. Two such structures will be called compatible if the sum of commutators is again a commutator. The natural class of compatible Poisson brackets is determined by effective symmetric Lie algebras dual in the sense of Cartan. Let the triplet (G, Ky a) be an effective symmetric Lie algebra and G = K Q P its decomposition under an involutive automorphism cr. Suppose that the representation ad iiC — End P is irreducible. 

The dualizing functor F Vectk —> Vectk mapping a vector space V to its dual V is a contravariant functor. Indeed, we recall from linear algebra that a linear transformation f V W induces a linear transformation / W —> V by mapping w G W to f w ) G V defined by... 

The set of all functions / 0 C) —> C is the dual of the vector space whose basis elements are indexed by the set 0 C). It is important to notice that the following equation describes the algebra representation of /(C) on this vector space ... 

In dual Lanczos transformation theory, we project onto a dynamically invariant subspace LZ or LZ called a diral Lanczos vector space. This projection is accomphshed with the projection operator... 

Given the starting vectors (ro and po), the remaining basis vectors for the dual Lanczos vector space LZ may be generated by means of the recursion relations... 

The dual Lanczos vector space generated with the starting vectors (rd and po) is finite dimensional when the Lanczos parameter vanishes for some finite value of N. For such cases, the dual Lanczos vector space is a dynamically invariant subspace of dimensionality N. This situation is realized when the operation (tm L or L pm) does not generate any new dynamical information, i.e., information not already contained in the vectors (r l x = 0, l and ft) 5 = 0,..., A - 1. ... 

One should bear in mind that a dual Lanczos vector space is a (fynamically invariant subspace by constraction, regardless of its dimensionality. 

The basis vectors for the dual Lanczos vector space may be generated with recursion relations identical inform to Eqs. (625) and (626). One only needs to replace Z with Z. As indicated earlier, the space LZ does not have to be generated when Z = Z. Even for cases not satisfying this syimnetry relation, we have not encountered a problem for which it is necessary to actually build ( z- Hence, we shall confine the remainder of our discussion to the dual Lanczos vector space LZ-... 

Lanczos vector space may be accomplished with a dual Lanczos transformation. More specifically,... 

In general, a dual Lanczos vector space depends on the choice of starting vectors (rd and po) used to build it. This is reflected in the observation that such a space does not, in general, include the whole domain of the operator L. More specifically, the operators L = Land L are not necessarily equivalent from a global point of view, i.e., in the space Q. Nonetheless, they are equivalent in the dual Lanczos vector space LZ-... 

In essence, a dual Lanczos transformation may be used to transform the retarded dynamics of a system into the closed retarded subdynamics of a dynamically invariant subspace. Of course, this dynamics is not necessarily equivalent to the retarded dynamics of the system. Nonetheless, knowledge of this subdynamics is sufficient for determining the properties of interest, provided we have properly biased the dual Lanczos vector space with the relevant dynamical information through our choice of (rol and po). 

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