Lectures on Real and Complex Vector Spaces

Summary Using Mizar [1], in the context of a real vector space, we introduce the concept of affine ratio of three aligned points (see [5]). It is also equivalent to the notion of “Mesure algèbrique”1, to the opposite of the notion of Teilverhältnis2 or to the opposite of the ordered length-ratio [9]. In the second part, we introduce the classic notion of “cross-ratio” of 4 points aligned in a real vector space. Finally, we show that if the real vector space is the real line, the notion corresponds to the classical notion3 [9]: The cross-ratio of a quadruple of distinct points on the real line with coordinates x1, x2, x3, x4 is given by: $$({x_1},{x_2};{x_3},{x_4}) = {{{x_3} - {x_1}} \over {{x_3} - {x_2}}}.{{{x_4} - {x_2}} \over {{x_4} - {x_1}}}$$ In the Mizar Mathematical Library, the vector spaces were first defined by Kusak, Leonczuk and Muzalewski in the article [6], while the actual real vector space was defined by Trybulec [10] and the complex vector space was defined by Endou [4]. Nakasho and Shidama have developed a solution to explore the notions introduced by different authors4 [7]. The definitions can be directly linked in the HTMLized version of the Mizar library5. The study of the cross-ratio will continue within the framework of the Klein- Beltrami model [2], [3]. For a generalized cross-ratio, see Papadopoulos [8].

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Complex Vector Spaces and General Inner Products

Elementary Linear Algebra ◽

10.1016/b978-0-12-374751-8.00006-8 ◽

2010 ◽

pp. 445-490

Author(s):

Stephen Andrilli ◽

David Hecker

Keyword(s):

Vector Spaces ◽

Complex Vector ◽

Inner Products

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Linear Algebra and the Dirac Notation

An Introduction to Quantum Computing ◽

10.1093/oso/9780198570004.003.0005 ◽

2006 ◽

Author(s):

Phillip Kaye ◽

Raymond Laflamme ◽

Michele Mosca

Keyword(s):

Hilbert Space ◽

Quantum Mechanics ◽

Quantum Computing ◽

Linear Algebra ◽

Hilbert Spaces ◽

Vector Spaces ◽

Complex Vector ◽

Dual Vector ◽

Finite Dimensional ◽

Basic Facts

We assume the reader has a strong background in elementary linear algebra. In this section we familiarize the reader with the algebraic notation used in quantum mechanics, remind the reader of some basic facts about complex vector spaces, and introduce some notions that might not have been covered in an elementary linear algebra course. The linear algebra notation used in quantum computing will likely be familiar to the student of physics, but may be alien to a student of mathematics or computer science. It is the Dirac notation, which was invented by Paul Dirac and which is used often in quantum mechanics. In mathematics and physics textbooks, vectors are often distinguished from scalars by writing an arrow over the identifying symbol: e.g a⃗. Sometimes boldface is used for this purpose: e.g. a. In the Dirac notation, the symbol identifying a vector is written inside a ‘ket’, and looks like |a⟩. We denote the dual vector for a (defined later) with a ‘bra’, written as ⟨a|. Then inner products will be written as ‘bra-kets’ (e.g. ⟨a|b⟩). We now carefully review the definitions of the main algebraic objects of interest, using the Dirac notation. The vector spaces we consider will be over the complex numbers, and are finite-dimensional, which significantly simplifies the mathematics we need. Such vector spaces are members of a class of vector spaces called Hilbert spaces. Nothing substantial is gained at this point by defining rigorously what a Hilbert space is, but virtually all the quantum computing literature refers to a finite-dimensional complex vector space by the name ‘Hilbert space’, and so we will follow this convention. We will use H to denote such a space. Since H is finite-dimensional, we can choose a basis and alternatively represent vectors (kets) in this basis as finite column vectors, and represent operators with finite matrices. As you see in Section 3, the Hilbert spaces of interest for quantum computing will typically have dimension 2n, for some positive integer n. This is because, as with classical information, we will construct larger state spaces by concatenating a string of smaller systems, usually of size two.

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A Purity Criterion for Pairs of Linear Transformations

Canadian Journal of Mathematics ◽

10.4153/cjm-1974-068-8 ◽

1974 ◽

Vol 26 (3) ◽

pp. 734-745 ◽

Cited By ~ 4

Author(s):

Uri Fixman ◽

Frank A. Zorzitto

Keyword(s):

Perturbation Methods ◽

Eigenvalue Problems ◽

Vector Spaces ◽

Infinite Dimension ◽

Linear Transformations ◽

Complex Vector ◽

Algebraic Investigation ◽

Image Position

In connection with the study of perturbation methods for differential eigenvalue problems, Aronszajn put forth a theory of systems (X, Y; A, B) consisting of a pair of linear transformations A, B:X → Y (see [1]; cf. also [2]). Here X and Y are complex vector spaces, possibly of infinite dimension. The algebraic aspects of this theory, where no restrictions of topological nature are imposed, where developed in [3] and [5]. We hasten to point out that the category of C2-systems (definition in § 1) in which this algebraic investigation takes place is equivalent to the category of all right modules over the ring of matrices of the form

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Purity And Copurity in Systems of Linear Transformations

Canadian Journal of Mathematics ◽

10.4153/cjm-1976-086-7 ◽

1976 ◽

Vol 28 (4) ◽

pp. 889-896

Author(s):

Frank Zorzitto

Keyword(s):

Vector Spaces ◽

Finite Codimension ◽

Linear Transformations ◽

Complex Vector ◽

Finite Dimensional ◽

Quotient System

Consider a system of N linear transformations A1, … , AN: V → W, where F and IF are complex vector spaces. Denote it for short by (F, W). A pair of subspaces X ⊂ V, Y ⊂ W such that determines a subsystem (X, Y) and a quotient system (V/X, W/Y) (with the induced transformations). The subsystem (X, Y) is of finite codimension in (V, W) if and only if V/X and W / Y are finite-dimensional. It is a direct summand of (V, W) in case there exist supplementary subspaces P of X in F and Q of F in IF such that (P, Q) is a subsystem.

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CR Mappings of Circular CR Manifolds

Canadian Mathematical Bulletin ◽

10.4153/cmb-1995-058-8 ◽

1995 ◽

Vol 38 (4) ◽

pp. 396-407

Author(s):

André Boivin ◽

Roman Dwilewicz

Keyword(s):

Finite Number ◽

Type Equation ◽

Vector Spaces ◽

Cr Manifold ◽

Complex Vector ◽

Cr Manifolds ◽

Cr Mappings

AbstractLet M be a circular CR manifold and let N be a rigid CR manifold in some complex vector spaces. The problem of the existence of local CR mappings from M into N is considered. Conditions are given which ensure that the space of such CR mappings depends on a finite number of parameters. The idea of the proof of the main result relies on a Bishop type equation for CR mappings. Roughly speaking, we look for CR mappings from M into N in the form F = (ƒ,g), we assume that g is given, then we find ƒ in terms of g and some parameters, and finally we look for conditions on g. It works independently of assumptions on the Levi forms of M and N, and there is also some freedom on the codimension of the manifolds.

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