Coordinate Transforms and Dual Bases: a Teaching Aid for Undergraduate Engineering Students
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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. -
Bornologically Isomorphic Representations of Tensor Distributions
Bornologically isomorphic representations of distributions on manifolds E. Nigsch Thursday 15th November, 2018 Abstract Distributional tensor fields can be regarded as multilinear mappings with distributional values or as (classical) tensor fields with distribu- tional coefficients. We show that the corresponding isomorphisms hold also in the bornological setting. 1 Introduction ′ ′ ′r s ′ Let D (M) := Γc(M, Vol(M)) and Ds (M) := Γc(M, Tr(M) ⊗ Vol(M)) be the strong duals of the space of compactly supported sections of the volume s bundle Vol(M) and of its tensor product with the tensor bundle Tr(M) over a manifold; these are the spaces of scalar and tensor distributions on M as defined in [?, ?]. A property of the space of tensor distributions which is fundamental in distributional geometry is given by the C∞(M)-module isomorphisms ′r ∼ s ′ ∼ r ′ Ds (M) = LC∞(M)(Tr (M), D (M)) = Ts (M) ⊗C∞(M) D (M) (1) (cf. [?, Theorem 3.1.12 and Corollary 3.1.15]) where C∞(M) is the space of smooth functions on M. In[?] a space of Colombeau-type nonlinear generalized tensor fields was constructed. This involved handling smooth functions (in the sense of convenient calculus as developed in [?]) in par- arXiv:1105.1642v1 [math.FA] 9 May 2011 ∞ r ′ ticular on the C (M)-module tensor products Ts (M) ⊗C∞(M) D (M) and Γ(E) ⊗C∞(M) Γ(F ), where Γ(E) denotes the space of smooth sections of a vector bundle E over M. In[?], however, only minor attention was paid to questions of topology on these tensor products. -
Lecture 4: April 8, 2021 1 Orthogonality and Orthonormality
Mathematical Toolkit Spring 2021 Lecture 4: April 8, 2021 Lecturer: Avrim Blum (notes based on notes from Madhur Tulsiani) 1 Orthogonality and orthonormality Definition 1.1 Two vectors u, v in an inner product space are said to be orthogonal if hu, vi = 0. A set of vectors S ⊆ V is said to consist of mutually orthogonal vectors if hu, vi = 0 for all u 6= v, u, v 2 S. A set of S ⊆ V is said to be orthonormal if hu, vi = 0 for all u 6= v, u, v 2 S and kuk = 1 for all u 2 S. Proposition 1.2 A set S ⊆ V n f0V g consisting of mutually orthogonal vectors is linearly inde- pendent. Proposition 1.3 (Gram-Schmidt orthogonalization) Given a finite set fv1,..., vng of linearly independent vectors, there exists a set of orthonormal vectors fw1,..., wng such that Span (fw1,..., wng) = Span (fv1,..., vng) . Proof: By induction. The case with one vector is trivial. Given the statement for k vectors and orthonormal fw1,..., wkg such that Span (fw1,..., wkg) = Span (fv1,..., vkg) , define k u + u = v − hw , v i · w and w = k 1 . k+1 k+1 ∑ i k+1 i k+1 k k i=1 uk+1 We can now check that the set fw1,..., wk+1g satisfies the required conditions. Unit length is clear, so let’s check orthogonality: k uk+1, wj = vk+1, wj − ∑ hwi, vk+1i · wi, wj = vk+1, wj − wj, vk+1 = 0. i=1 Corollary 1.4 Every finite dimensional inner product space has an orthonormal basis. -
Solution: the First Element of the Dual Basis Is the Linear Function Α
Homework assignment 7 pp. 105 3 Exercise 2. Let B = f®1; ®2; ®3g be the basis for C defined by ®1 = (1; 0; ¡1) ®2 = (1; 1; 1) ®3 = (2; 2; 0): Find the dual basis of B. Solution: ¤ The first element of the dual basis is the linear function ®1 such ¤ ¤ ¤ that ®1(®1) = 1; ®1(®2) = 0 and ®1(®3) = 0. To describe such a function more explicitly we need to find its values on the standard basis vectors e1, e2 and e3. To do this express e1; e2; e3 through ®1; ®2; ®3 (refer to the solution of Exercise 1 pp. 54-55 from Homework 6). For each i = 1; 2; 3 you will find the numbers ai; bi; ci such that e1 = ai®1 + bi®2 + ci®3 (i.e. the coordinates of ei relative to the ¤ ¤ ¤ basis ®1; ®2; ®3). Then by linearity of ®1 we get that ®1(ei) = ai. Then ®2(ei) = bi, ¤ and ®3(ei) = ci. This is the answer. It can also be reformulated as follows. If P is the transition matrix from the standard basis e1; e2; e3 to ®1; ®2; ®3, i.e. ¡1 t (®1; ®2; ®3) = (e1; e2; e3)P , then (P ) is the transition matrix from the dual basis ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¡1 t e1; e2; e3 to the dual basis ®1; ®2; a3, i.e. (®1; ®2; a3) = (e1; e2; e3)(P ) . Note that this problem is basically the change of coordinates problem: e.g. ¤ 3 the value of ®1 on the vector v 2 C is the first coordinate of v relative to the basis ®1; ®2; ®3. -
Orthonormality and the Gram-Schmidt Process
Unit 4, Section 2: The Gram-Schmidt Process Orthonormality and the Gram-Schmidt Process The basis (e1; e2; : : : ; en) n n for R (or C ) is considered the standard basis for the space because of its geometric properties under the standard inner product: 1. jjeijj = 1 for all i, and 2. ei, ej are orthogonal whenever i 6= j. With this idea in mind, we record the following definitions: Definitions 6.23/6.25. Let V be an inner product space. • A list of vectors in V is called orthonormal if each vector in the list has norm 1, and if each pair of distinct vectors is orthogonal. • A basis for V is called an orthonormal basis if the basis is an orthonormal list. Remark. If a list (v1; : : : ; vn) is orthonormal, then ( 0 if i 6= j hvi; vji = 1 if i = j: Example. The list (e1; e2; : : : ; en) n n forms an orthonormal basis for R =C under the standard inner products on those spaces. 2 Example. The standard basis for Mn(C) consists of n matrices eij, 1 ≤ i; j ≤ n, where eij is the n × n matrix with a 1 in the ij entry and 0s elsewhere. Under the standard inner product on Mn(C) this is an orthonormal basis for Mn(C): 1. heij; eiji: ∗ heij; eiji = tr (eijeij) = tr (ejieij) = tr (ejj) = 1: 1 Unit 4, Section 2: The Gram-Schmidt Process 2. heij; ekli, k 6= i or j 6= l: ∗ heij; ekli = tr (ekleij) = tr (elkeij) = tr (0) if k 6= i, or tr (elj) if k = i but l 6= j = 0: So every vector in the list has norm 1, and every distinct pair of vectors is orthogonal. -
Vectors and Dual Vectors
Vectors By now you have a pretty good experience with \vectors". Usually, a vector is defined as a quantity that has a direction and a magnitude, such as a position vector, velocity vector, acceleration vector, etc. However, the notion of a vector has a considerably wider realm of applicability than these examples might suggest. The set of all real numbers forms a vector space, as does the set of all complex numbers. The set of functions on a set (e.g., functions of one variable, f(x)) form a vector space. Solutions of linear homogeneous equations form a vector space. We begin by giving the abstract rules for forming a space of vectors, also known as a vector space. A vector space V is a set equipped with an operation of \addition" and an additive identity. The elements of the set are called vectors, which we shall denote as ~u, ~v, ~w, etc. For now, you can think of them as position vectors in order to keep yourself sane. Addition, is an operation in which two vectors, say ~u and ~v, can be combined to make another vector, say, ~w. We denote this operation by the symbol \+": ~u + ~v = ~w: (1) Do not be fooled by this simple notation. The \addition" of vectors may be quite a different operation than ordinary arithmetic addition. For example, if we view position vectors in the x-y plane as \arrows" drawn from the origin, the addition of vectors is defined by the parallelogram rule. Clearly this rule is quite different than ordinary \addition". -
Duality, Part 1: Dual Bases and Dual Maps Notation
Duality, part 1: Dual Bases and Dual Maps Notation F denotes either R or C. V and W denote vector spaces over F. Define ': R3 ! R by '(x; y; z) = 4x − 5y + 2z. Then ' is a linear functional on R3. n n Fix (b1;:::; bn) 2 C . Define ': C ! C by '(z1;:::; zn) = b1z1 + ··· + bnzn: Then ' is a linear functional on Cn. Define ': P(R) ! R by '(p) = 3p00(5) + 7p(4). Then ' is a linear functional on P(R). R 1 Define ': P(R) ! R by '(p) = 0 p(x) dx. Then ' is a linear functional on P(R). Examples: Linear Functionals Definition: linear functional A linear functional on V is a linear map from V to F. In other words, a linear functional is an element of L(V; F). n n Fix (b1;:::; bn) 2 C . Define ': C ! C by '(z1;:::; zn) = b1z1 + ··· + bnzn: Then ' is a linear functional on Cn. Define ': P(R) ! R by '(p) = 3p00(5) + 7p(4). Then ' is a linear functional on P(R). R 1 Define ': P(R) ! R by '(p) = 0 p(x) dx. Then ' is a linear functional on P(R). Linear Functionals Definition: linear functional A linear functional on V is a linear map from V to F. In other words, a linear functional is an element of L(V; F). Examples: Define ': R3 ! R by '(x; y; z) = 4x − 5y + 2z. Then ' is a linear functional on R3. Define ': P(R) ! R by '(p) = 3p00(5) + 7p(4). Then ' is a linear functional on P(R). R 1 Define ': P(R) ! R by '(p) = 0 p(x) dx. -
A Some Basic Rules of Tensor Calculus
A Some Basic Rules of Tensor Calculus The tensor calculus is a powerful tool for the description of the fundamentals in con- tinuum mechanics and the derivation of the governing equations for applied prob- lems. In general, there are two possibilities for the representation of the tensors and the tensorial equations: – the direct (symbolic) notation and – the index (component) notation The direct notation operates with scalars, vectors and tensors as physical objects defined in the three dimensional space. A vector (first rank tensor) a is considered as a directed line segment rather than a triple of numbers (coordinates). A second rank tensor A is any finite sum of ordered vector pairs A = a b + ... +c d. The scalars, vectors and tensors are handled as invariant (independent⊗ from the choice⊗ of the coordinate system) objects. This is the reason for the use of the direct notation in the modern literature of mechanics and rheology, e.g. [29, 32, 49, 123, 131, 199, 246, 313, 334] among others. The index notation deals with components or coordinates of vectors and tensors. For a selected basis, e.g. gi, i = 1, 2, 3 one can write a = aig , A = aibj + ... + cidj g g i i ⊗ j Here the Einstein’s summation convention is used: in one expression the twice re- peated indices are summed up from 1 to 3, e.g. 3 3 k k ik ik a gk ∑ a gk, A bk ∑ A bk ≡ k=1 ≡ k=1 In the above examples k is a so-called dummy index. Within the index notation the basic operations with tensors are defined with respect to their coordinates, e. -
Multilinear Algebra
Appendix A Multilinear Algebra This chapter presents concepts from multilinear algebra based on the basic properties of finite dimensional vector spaces and linear maps. The primary aim of the chapter is to give a concise introduction to alternating tensors which are necessary to define differential forms on manifolds. Many of the stated definitions and propositions can be found in Lee [1], Chaps. 11, 12 and 14. Some definitions and propositions are complemented by short and simple examples. First, in Sect. A.1 dual and bidual vector spaces are discussed. Subsequently, in Sects. A.2–A.4, tensors and alternating tensors together with operations such as the tensor and wedge product are introduced. Lastly, in Sect. A.5, the concepts which are necessary to introduce the wedge product are summarized in eight steps. A.1 The Dual Space Let V be a real vector space of finite dimension dim V = n.Let(e1,...,en) be a basis of V . Then every v ∈ V can be uniquely represented as a linear combination i v = v ei , (A.1) where summation convention over repeated indices is applied. The coefficients vi ∈ R arereferredtoascomponents of the vector v. Throughout the whole chapter, only finite dimensional real vector spaces, typically denoted by V , are treated. When not stated differently, summation convention is applied. Definition A.1 (Dual Space)Thedual space of V is the set of real-valued linear functionals ∗ V := {ω : V → R : ω linear} . (A.2) The elements of the dual space V ∗ are called linear forms on V . © Springer International Publishing Switzerland 2015 123 S.R. -
Inner Product Spaces
CHAPTER 6 Woman teaching geometry, from a fourteenth-century edition of Euclid’s geometry book. Inner Product Spaces In making the definition of a vector space, we generalized the linear structure (addition and scalar multiplication) of R2 and R3. We ignored other important features, such as the notions of length and angle. These ideas are embedded in the concept we now investigate, inner products. Our standing assumptions are as follows: 6.1 Notation F, V F denotes R or C. V denotes a vector space over F. LEARNING OBJECTIVES FOR THIS CHAPTER Cauchy–Schwarz Inequality Gram–Schmidt Procedure linear functionals on inner product spaces calculating minimum distance to a subspace Linear Algebra Done Right, third edition, by Sheldon Axler 164 CHAPTER 6 Inner Product Spaces 6.A Inner Products and Norms Inner Products To motivate the concept of inner prod- 2 3 x1 , x 2 uct, think of vectors in R and R as x arrows with initial point at the origin. x R2 R3 H L The length of a vector in or is called the norm of x, denoted x . 2 k k Thus for x .x1; x2/ R , we have The length of this vector x is p D2 2 2 x x1 x2 . p 2 2 x1 x2 . k k D C 3 C Similarly, if x .x1; x2; x3/ R , p 2D 2 2 2 then x x1 x2 x3 . k k D C C Even though we cannot draw pictures in higher dimensions, the gener- n n alization to R is obvious: we define the norm of x .x1; : : : ; xn/ R D 2 by p 2 2 x x1 xn : k k D C C The norm is not linear on Rn. -
Geometric Algebra Techniques for General Relativity
Geometric Algebra Techniques for General Relativity Matthew R. Francis∗ and Arthur Kosowsky† Dept. of Physics and Astronomy, Rutgers University 136 Frelinghuysen Road, Piscataway, NJ 08854 (Dated: February 4, 2008) Geometric (Clifford) algebra provides an efficient mathematical language for describing physical problems. We formulate general relativity in this language. The resulting formalism combines the efficiency of differential forms with the straightforwardness of coordinate methods. We focus our attention on orthonormal frames and the associated connection bivector, using them to find the Schwarzschild and Kerr solutions, along with a detailed exposition of the Petrov types for the Weyl tensor. PACS numbers: 02.40.-k; 04.20.Cv Keywords: General relativity; Clifford algebras; solution techniques I. INTRODUCTION Geometric (or Clifford) algebra provides a simple and natural language for describing geometric concepts, a point which has been argued persuasively by Hestenes [1] and Lounesto [2] among many others. Geometric algebra (GA) unifies many other mathematical formalisms describing specific aspects of geometry, including complex variables, matrix algebra, projective geometry, and differential geometry. Gravitation, which is usually viewed as a geometric theory, is a natural candidate for translation into the language of geometric algebra. This has been done for some aspects of gravitational theory; notably, Hestenes and Sobczyk have shown how geometric algebra greatly simplifies certain calculations involving the curvature tensor and provides techniques for classifying the Weyl tensor [3, 4]. Lasenby, Doran, and Gull [5] have also discussed gravitation using geometric algebra via a reformulation in terms of a gauge principle. In this paper, we formulate standard general relativity in terms of geometric algebra. A comprehensive overview like the one presented here has not previously appeared in the literature, although unpublished works of Hestenes and of Doran take significant steps in this direction. -
Inner Product Spaces Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (March 2, 2007)
MAT067 University of California, Davis Winter 2007 Inner Product Spaces Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (March 2, 2007) The abstract definition of vector spaces only takes into account algebraic properties for the addition and scalar multiplication of vectors. For vectors in Rn, for example, we also have geometric intuition which involves the length of vectors or angles between vectors. In this section we discuss inner product spaces, which are vector spaces with an inner product defined on them, which allow us to introduce the notion of length (or norm) of vectors and concepts such as orthogonality. 1 Inner product In this section V is a finite-dimensional, nonzero vector space over F. Definition 1. An inner product on V is a map ·, · : V × V → F (u, v) →u, v with the following properties: 1. Linearity in first slot: u + v, w = u, w + v, w for all u, v, w ∈ V and au, v = au, v; 2. Positivity: v, v≥0 for all v ∈ V ; 3. Positive definiteness: v, v =0ifandonlyifv =0; 4. Conjugate symmetry: u, v = v, u for all u, v ∈ V . Remark 1. Recall that every real number x ∈ R equals its complex conjugate. Hence for real vector spaces the condition about conjugate symmetry becomes symmetry. Definition 2. An inner product space is a vector space over F together with an inner product ·, ·. Copyright c 2007 by the authors. These lecture notes may be reproduced in their entirety for non- commercial purposes. 2NORMS 2 Example 1. V = Fn n u =(u1,...,un),v =(v1,...,vn) ∈ F Then n u, v = uivi.