Inner Product Spaces and Orthogonality
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The Grassmann Manifold
The Grassmann Manifold 1. For vector spaces V and W denote by L(V; W ) the vector space of linear maps from V to W . Thus L(Rk; Rn) may be identified with the space Rk£n of k £ n matrices. An injective linear map u : Rk ! V is called a k-frame in V . The set k n GFk;n = fu 2 L(R ; R ): rank(u) = kg of k-frames in Rn is called the Stiefel manifold. Note that the special case k = n is the general linear group: k k GLk = fa 2 L(R ; R ) : det(a) 6= 0g: The set of all k-dimensional (vector) subspaces ¸ ½ Rn is called the Grassmann n manifold of k-planes in R and denoted by GRk;n or sometimes GRk;n(R) or n GRk(R ). Let k ¼ : GFk;n ! GRk;n; ¼(u) = u(R ) denote the map which assigns to each k-frame u the subspace u(Rk) it spans. ¡1 For ¸ 2 GRk;n the fiber (preimage) ¼ (¸) consists of those k-frames which form a basis for the subspace ¸, i.e. for any u 2 ¼¡1(¸) we have ¡1 ¼ (¸) = fu ± a : a 2 GLkg: Hence we can (and will) view GRk;n as the orbit space of the group action GFk;n £ GLk ! GFk;n :(u; a) 7! u ± a: The exercises below will prove the following n£k Theorem 2. The Stiefel manifold GFk;n is an open subset of the set R of all n £ k matrices. There is a unique differentiable structure on the Grassmann manifold GRk;n such that the map ¼ is a submersion. -
Introduction to Linear Bialgebra
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of New Mexico University of New Mexico UNM Digital Repository Mathematics and Statistics Faculty and Staff Publications Academic Department Resources 2005 INTRODUCTION TO LINEAR BIALGEBRA Florentin Smarandache University of New Mexico, [email protected] W.B. Vasantha Kandasamy K. Ilanthenral Follow this and additional works at: https://digitalrepository.unm.edu/math_fsp Part of the Algebra Commons, Analysis Commons, Discrete Mathematics and Combinatorics Commons, and the Other Mathematics Commons Recommended Citation Smarandache, Florentin; W.B. Vasantha Kandasamy; and K. Ilanthenral. "INTRODUCTION TO LINEAR BIALGEBRA." (2005). https://digitalrepository.unm.edu/math_fsp/232 This Book is brought to you for free and open access by the Academic Department Resources at UNM Digital Repository. It has been accepted for inclusion in Mathematics and Statistics Faculty and Staff Publications by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected], [email protected], [email protected]. INTRODUCTION TO LINEAR BIALGEBRA W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600036, India e-mail: [email protected] web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gallup, NM 87301, USA e-mail: [email protected] K. Ilanthenral Editor, Maths Tiger, Quarterly Journal Flat No.11, Mayura Park, 16, Kazhikundram Main Road, Tharamani, Chennai – 600 113, India e-mail: [email protected] HEXIS Phoenix, Arizona 2005 1 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. -
A New Description of Space and Time Using Clifford Multivectors
A new description of space and time using Clifford multivectors James M. Chappell† , Nicolangelo Iannella† , Azhar Iqbal† , Mark Chappell‡ , Derek Abbott† †School of Electrical and Electronic Engineering, University of Adelaide, South Australia 5005, Australia ‡Griffith Institute, Griffith University, Queensland 4122, Australia Abstract Following the development of the special theory of relativity in 1905, Minkowski pro- posed a unified space and time structure consisting of three space dimensions and one time dimension, with relativistic effects then being natural consequences of this space- time geometry. In this paper, we illustrate how Clifford’s geometric algebra that utilizes multivectors to represent spacetime, provides an elegant mathematical framework for the study of relativistic phenomena. We show, with several examples, how the application of geometric algebra leads to the correct relativistic description of the physical phenomena being considered. This approach not only provides a compact mathematical representa- tion to tackle such phenomena, but also suggests some novel insights into the nature of time. Keywords: Geometric algebra, Clifford space, Spacetime, Multivectors, Algebraic framework 1. Introduction The physical world, based on early investigations, was deemed to possess three inde- pendent freedoms of translation, referred to as the three dimensions of space. This naive conclusion is also supported by more sophisticated analysis such as the existence of only five regular polyhedra and the inverse square force laws. If we lived in a world with four spatial dimensions, for example, we would be able to construct six regular solids, and in arXiv:1205.5195v2 [math-ph] 11 Oct 2012 five dimensions and above we would find only three [1]. -
21. Orthonormal Bases
21. Orthonormal Bases The canonical/standard basis 011 001 001 B C B C B C B0C B1C B0C e1 = B.C ; e2 = B.C ; : : : ; en = B.C B.C B.C B.C @.A @.A @.A 0 0 1 has many useful properties. • Each of the standard basis vectors has unit length: q p T jjeijj = ei ei = ei ei = 1: • The standard basis vectors are orthogonal (in other words, at right angles or perpendicular). T ei ej = ei ej = 0 when i 6= j This is summarized by ( 1 i = j eT e = δ = ; i j ij 0 i 6= j where δij is the Kronecker delta. Notice that the Kronecker delta gives the entries of the identity matrix. Given column vectors v and w, we have seen that the dot product v w is the same as the matrix multiplication vT w. This is the inner product on n T R . We can also form the outer product vw , which gives a square matrix. 1 The outer product on the standard basis vectors is interesting. Set T Π1 = e1e1 011 B C B0C = B.C 1 0 ::: 0 B.C @.A 0 01 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 0 . T Πn = enen 001 B C B0C = B.C 0 0 ::: 1 B.C @.A 1 00 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 1 In short, Πi is the diagonal square matrix with a 1 in the ith diagonal position and zeros everywhere else. -
Orthogonal Bases and the -So in Section 4.8 We Discussed the Problem of Finding the Orthogonal Projection P
Orthogonal Bases and the -So In Section 4.8 we discussed the problemR. of finding the orthogonal projection p the vector b into the V of , . , suhspace the vectors , v2 ho If v1 v,, form a for V, and the in x n matrix A has these basis vectors as its column vectors. ilt the orthogonal projection p is given by p = Ax where x is the (unique) solution of the normal system ATAx = A7b. The formula for p takes an especially simple and attractive Form when the ba vectors , . .. , v1 v are mutually orthogonal. DEFINITION Orthogonal Basis An orthogonal basis for the suhspacc V of R” is a basis consisting of vectors , ,v,, that are mutually orthogonal, so that v v = 0 if I j. If in additii these basis vectors are unit vectors, so that v1 . = I for i = 1. 2 n, thct the orthogonal basis is called an orthonormal basis. Example 1 The vectors = (1, 1,0), v2 = (1, —1,2), v3 = (—1,1,1) form an orthogonal basis for . We “normalize” ‘ R3 can this orthogonal basis 1w viding each basis vector by its length: If w1=—- (1=1,2,3), lvii 4.9 Orthogonal Bases and the Gram-Schmidt Algorithm 295 then the vectors /1 I 1 /1 1 ‘\ 1 2” / I w1 0) W2 = —— _z) W3 for . form an orthonormal basis R3 , . ..., v, of the m x ii matrix A Now suppose that the column vectors v v2 form an orthogonal basis for the suhspacc V of R’. Then V}.VI 0 0 v2.v .. -
Determinant Notes
68 UNIT FIVE DETERMINANTS 5.1 INTRODUCTION In unit one the determinant of a 2× 2 matrix was introduced and used in the evaluation of a cross product. In this chapter we extend the definition of a determinant to any size square matrix. The determinant has a variety of applications. The value of the determinant of a square matrix A can be used to determine whether A is invertible or noninvertible. An explicit formula for A–1 exists that involves the determinant of A. Some systems of linear equations have solutions that can be expressed in terms of determinants. 5.2 DEFINITION OF THE DETERMINANT a11 a12 Recall that in chapter one the determinant of the 2× 2 matrix A = was a21 a22 defined to be the number a11a22 − a12 a21 and that the notation det (A) or A was used to represent the determinant of A. For any given n × n matrix A = a , the notation A [ ij ]n×n ij will be used to denote the (n −1)× (n −1) submatrix obtained from A by deleting the ith row and the jth column of A. The determinant of any size square matrix A = a is [ ij ]n×n defined recursively as follows. Definition of the Determinant Let A = a be an n × n matrix. [ ij ]n×n (1) If n = 1, that is A = [a11], then we define det (A) = a11 . n 1k+ (2) If na>=1, we define det(A) ∑(-1)11kk det(A ) k=1 Example If A = []5 , then by part (1) of the definition of the determinant, det (A) = 5. -
2 Hilbert Spaces You Should Have Seen Some Examples Last Semester
2 Hilbert spaces You should have seen some examples last semester. The simplest (finite-dimensional) ex- C • A complex Hilbert space H is a complete normed space over whose norm is derived from an ample is Cn with its standard inner product. It’s worth recalling from linear algebra that if V is inner product. That is, we assume that there is a sesquilinear form ( , ): H H C, linear in · · × → an n-dimensional (complex) vector space, then from any set of n linearly independent vectors we the first variable and conjugate linear in the second, such that can manufacture an orthonormal basis e1, e2,..., en using the Gram-Schmidt process. In terms of this basis we can write any v V in the form (f ,д) = (д, f ), ∈ v = a e , a = (v, e ) (f , f ) 0 f H, and (f , f ) = 0 = f = 0. i i i i ≥ ∀ ∈ ⇒ ∑ The norm and inner product are related by which can be derived by taking the inner product of the equation v = aiei with ei. We also have n ∑ (f , f ) = f 2. ∥ ∥ v 2 = a 2. ∥ ∥ | i | i=1 We will always assume that H is separable (has a countable dense subset). ∑ Standard infinite-dimensional examples are l2(N) or l2(Z), the space of square-summable As usual for a normed space, the distance on H is given byd(f ,д) = f д = (f д, f д). • • ∥ − ∥ − − sequences, and L2(Ω) where Ω is a measurable subset of Rn. The Cauchy-Schwarz and triangle inequalities, • √ (f ,д) f д , f + д f + д , | | ≤ ∥ ∥∥ ∥ ∥ ∥ ≤ ∥ ∥ ∥ ∥ 2.1 Orthogonality can be derived fairly easily from the inner product. -
Orthonormal Bases in Hilbert Space APPM 5440 Fall 2017 Applied Analysis
Orthonormal Bases in Hilbert Space APPM 5440 Fall 2017 Applied Analysis Stephen Becker November 18, 2017(v4); v1 Nov 17 2014 and v2 Jan 21 2016 Supplementary notes to our textbook (Hunter and Nachtergaele). These notes generally follow Kreyszig’s book. The reason for these notes is that this is a simpler treatment that is easier to follow; the simplicity is because we generally do not consider uncountable nets, but rather only sequences (which are countable). I have tried to identify the theorems from both books; theorems/lemmas with numbers like 3.3-7 are from Kreyszig. Note: there are two versions of these notes, with and without proofs. The version without proofs will be distributed to the class initially, and the version with proofs will be distributed later (after any potential homework assignments, since some of the proofs maybe assigned as homework). This version does not have proofs. 1 Basic Definitions NOTE: Kreyszig defines inner products to be linear in the first term, and conjugate linear in the second term, so hαx, yi = αhx, yi = hx, αy¯ i. In contrast, Hunter and Nachtergaele define the inner product to be linear in the second term, and conjugate linear in the first term, so hαx,¯ yi = αhx, yi = hx, αyi. We will follow the convention of Hunter and Nachtergaele, so I have re-written the theorems from Kreyszig accordingly whenever the order is important. Of course when working with the real field, order is completely unimportant. Let X be an inner product space. Then we can define a norm on X by kxk = phx, xi, x ∈ X. -
THE DIMENSION of a VECTOR SPACE 1. Introduction This Handout
THE DIMENSION OF A VECTOR SPACE KEITH CONRAD 1. Introduction This handout is a supplementary discussion leading up to the definition of dimension of a vector space and some of its properties. We start by defining the span of a finite set of vectors and linear independence of a finite set of vectors, which are combined to define the all-important concept of a basis. Definition 1.1. Let V be a vector space over a field F . For any finite subset fv1; : : : ; vng of V , its span is the set of all of its linear combinations: Span(v1; : : : ; vn) = fc1v1 + ··· + cnvn : ci 2 F g: Example 1.2. In F 3, Span((1; 0; 0); (0; 1; 0)) is the xy-plane in F 3. Example 1.3. If v is a single vector in V then Span(v) = fcv : c 2 F g = F v is the set of scalar multiples of v, which for nonzero v should be thought of geometrically as a line (through the origin, since it includes 0 · v = 0). Since sums of linear combinations are linear combinations and the scalar multiple of a linear combination is a linear combination, Span(v1; : : : ; vn) is a subspace of V . It may not be all of V , of course. Definition 1.4. If fv1; : : : ; vng satisfies Span(fv1; : : : ; vng) = V , that is, if every vector in V is a linear combination from fv1; : : : ; vng, then we say this set spans V or it is a spanning set for V . Example 1.5. In F 2, the set f(1; 0); (0; 1); (1; 1)g is a spanning set of F 2. -
On Isometries on Some Banach Spaces: Part I
Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections On isometries on some Banach spaces { something old, something new, something borrowed, something blue, Part I Dijana Iliˇsevi´c University of Zagreb, Croatia Recent Trends in Operator Theory and Applications Memphis, TN, USA, May 3{5, 2018 Recent work of D.I. has been fully supported by the Croatian Science Foundation under the project IP-2016-06-1046. Dijana Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections Something old, something new, something borrowed, something blue is referred to the collection of items that helps to guarantee fertility and prosperity. Dijana Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections Isometries Isometries are maps between metric spaces which preserve distance between elements. Definition Let (X ; j · j) and (Y; k · k) be two normed spaces over the same field. A linear map ': X!Y is called a linear isometry if k'(x)k = jxj; x 2 X : We shall be interested in surjective linear isometries on Banach spaces. One of the main problems is to give explicit description of isometries on a particular space. Dijana Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections Isometries Isometries are maps between metric spaces which preserve distance between elements. Definition Let (X ; j · j) and (Y; k · k) be two normed spaces over the same field. -
Determinants Math 122 Calculus III D Joyce, Fall 2012
Determinants Math 122 Calculus III D Joyce, Fall 2012 What they are. A determinant is a value associated to a square array of numbers, that square array being called a square matrix. For example, here are determinants of a general 2 × 2 matrix and a general 3 × 3 matrix. a b = ad − bc: c d a b c d e f = aei + bfg + cdh − ceg − afh − bdi: g h i The determinant of a matrix A is usually denoted jAj or det (A). You can think of the rows of the determinant as being vectors. For the 3×3 matrix above, the vectors are u = (a; b; c), v = (d; e; f), and w = (g; h; i). Then the determinant is a value associated to n vectors in Rn. There's a general definition for n×n determinants. It's a particular signed sum of products of n entries in the matrix where each product is of one entry in each row and column. The two ways you can choose one entry in each row and column of the 2 × 2 matrix give you the two products ad and bc. There are six ways of chosing one entry in each row and column in a 3 × 3 matrix, and generally, there are n! ways in an n × n matrix. Thus, the determinant of a 4 × 4 matrix is the signed sum of 24, which is 4!, terms. In this general definition, half the terms are taken positively and half negatively. In class, we briefly saw how the signs are determined by permutations. -
Computing the Complexity of the Relation of Isometry Between Separable Banach Spaces
Computing the complexity of the relation of isometry between separable Banach spaces Julien Melleray Abstract We compute here the Borel complexity of the relation of isometry between separable Banach spaces, using results of Gao, Kechris [1] and Mayer-Wolf [4]. We show that this relation is Borel bireducible to the universal relation for Borel actions of Polish groups. 1 Introduction Over the past fteen years or so, the theory of complexity of Borel equivalence relations has been a very active eld of research; in this paper, we compute the complexity of a relation of geometric nature, the relation of (linear) isom- etry between separable Banach spaces. Before stating precisely our result, we begin by quickly recalling the basic facts and denitions that we need in the following of the article; we refer the reader to [3] for a thorough introduction to the concepts and methods of descriptive set theory. Before going on with the proof and denitions, it is worth pointing out that this article mostly consists in putting together various results which were already known, and deduce from them after an easy manipulation the complexity of the relation of isometry between separable Banach spaces. Since this problem has been open for a rather long time, it still seems worth it to explain how this can be done, and give pointers to the literature. Still, it seems to the author that this will be mostly of interest to people with a knowledge of descriptive set theory, so below we don't recall descriptive set-theoretic notions but explain quickly what Lipschitz-free Banach spaces are and how that theory works.