DOCSLIB.ORG

Basic Multivector Calculus, Proceedings of FAN 2008, Hiroshima, Japan, 23+24 October 2008

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Basic Multivector Calculus, Proceedings of FAN 2008, Hiroshima, Japan, 23+24 October 2008 E. Hitzer, Basic Multivector Calculus, Proceedings of FAN 2008, Hiroshima, Japan, 23+24 October 2008. Basic Multivector Calculus Eckhard Hitzer, Department of Applied Physics, Faculty of Engineering, University of Fukui Abstract: We begin with introducing the generalization of real, complex, and quaternion numbers to hypercomplex numbers, also known as Clifford numbers, or multivectors of geometric algebra. Multivectors encode everything from vectors, rotations, scaling transformations, improper transformations (reflections, inversions), geometric objects (like lines and spheres), spinors, and tensors, and the like. Multivector calculus allows to define functions mapping multivectors to multivectors, differentiation, integration, function norms, multivector Fourier transformations and wavelet transformations, filtering, windowing, etc. We give a basic introduction into this general mathematical language, which has fascinating applications in physics, engineering, and computer science. more didactic approach for newcomers to geometric algebra. 1. Introduction G(I) is the full geometric algebra over all vectors in the “Now faith is being sure of what we hope for and certain of what we do not see. This is what the ancients were n-dimensional unit pseudoscalar I e1 e2 ... en . commended for. By faith we understand that the universe was formed at God’s command, so that what is seen was not made 1 A G (I) is the n-dimensional vector sub-space of out of what was visible.” [7] n The German 19th century mathematician H. Grassmann had grade-1 elements in G(I) spanned by e1,e2 ,...,en . For the clear vision, that his “extension theory (now developed to geometric calculus) … forms the keystone of the entire 1 vectors a,b,c An G (I) and scalars , ,, ; structure of mathematics.”[6] The algebraic “grammar” of this universal form of calculus is geometric algebra (or Clifford G(I) has the fundamental properties of algebra). That geometric calculus is a truly unifying approach associativity to all of calculus will be demonstrated here by developing some basics of the vector differential calculus part of a(bc) (ab)c, a (b c) (a b) c, (1) geometric calculus. The basic geometric algebra[1,2] necessary for this is commutativity a a, a b b a, (2) compiled in section 2. Then section 3 develops vector distributivity differential calculus with the help of few simple definitions. This approach is generically coordinate free, and fully shows a(b c) ab ac, (b c)a ba ca, (3) both the concrete and abstract geometric and algebraic beauty of the “keystone” of mathematics. The full generalization to linearity (a b) a b (a b), (4) multivector calculus is shown in [2,10], applications in [11]. scalar square (vector length ) a 2. Basic Geometric Algebra 2 2 This section is a basic summary of important relationships in a aa a a a . (5) geometric algebra. This summary mainly serves as a reference section for the vector differential calculus to be developed in The geometric product ab is related to the (scalar) inner the following section. Most of the relationships listed here are to be found in the synopsis of geometric algebra and in product a b and to the (bivector or 2-vector) outer product chapters 1 and 2 of [1], as well as in chapter 1 of [2], together with relevant proofs. Beyond that [1] and [2] follow a much a b by ab a b a b , (6) a(A B) aA aB . (19) with The inner and outer products of homogeneous multivectors 1 (13) A and B are defined ([2], p. 6, (1.21), (1.22)) as a b (ab ba) b a ab a b ab , (7) r s 2 0 (13) A B A B for r,s 0 , (20) 1 r s r s r s a b (ab ba) b a ab a b ab (8) 2 2 A B 0 for r 0 or s 0 , (21) The inner and the outer product are both linear and distributive r s A B A B , (22) a (b c) a b a c , (9) r s r s r s a (b c) a b a c . (10) Ar Ar Ar for scalar . (23) A unit vector aˆ in the direction of a is The inner (and outer) product is again linear and distributive a 2 (A ) B A (B ) (A B ) A B (24) aˆ , with aˆ aˆaˆ 1, a aˆ a . (11) r s r s r s r s a A (B C ) A B A C , (25) The inverse of a vector is r s t r s r t 1 1 a a aˆ . (26) a . (12) (Bs Ct ) Bs Ct a a2 a 2 a The reverse of a multivector is A multivector A can be uniquely decomposed into its ~ n A (1)k (k1) / 2 A . (27) homogeneous grade k parts ( grade k selector): k k k1 A A A A ... A ... A (13) [2] uses a dagger instead of the tilde. 0 1 2 k n Special examples are scalar vector bivector k vector pseudo scalar ~ ~ ~ , a a , (a b ) b a a b , …(28) If A is homogeneous of grade k one often simply writes A A A . (14) The scalar magnitude A of a multivector A is k k Grade selection is invariant under scalar multiplication n 2 ~ ~ A A A A 2 A A , (29) A A . (15) 0 r r k k scalar product r1 The consistent definition of inner and outer products of 2 where the separate term A is in particular due to the 0 vectors a and r-vectors A is r definition of the inner product in [2], p. 6, (1.21). The magnitude allows to define the inverse for simple k-blade 1 r a A aA (aA (1) A a), (16) vectors r r r 1 2 r r ~ 1 A 1 1 1 r , with . (30) a A aA (aA (1) A a) (17) A 2 A A AA 1 r r r 1 2 r r A By linearity the full geometric product of a vector and a Alternative ways to express a A G1(I) are multivector A is then n aA a A a A . (18) I a 0 or Ia I a . (31) This extends to the distributive multiplication with arbitrary The projection of a into A G1(I) is multivectors A, B n n n (a b) (c d) a (b (c d)) P (a) P(a) ak a a a ak a, (32) (42) I k k k 1 k 1 (a d)(b c) (a c)(b d ), where ak is the reciprocal frame defined by (a b)2 (a b) (a b) (a b)2 a2b 2 2 if (b a) (a b) a b , (43) k k 1 j k a a j j Kronecker delta . (33) 0 if j k and the Jacobi identity A general convention is that inner products a b and a (b c) b (c a) c (a b) 0. (44) The commutator product of multivectors A,B is outer products a b have priority over geometric products 1 A B (AB BA). (45) ab , e.g. 2 One useful identity using it is a bc de (a b)(c d )e . (34) (a b) A ab A A ab ab A A ab.(46) The The projection of a multivector B on a subspace described by a commutator product is to be distinguished from the cross simple m-vector (m-blade) A a a ... a ,m n m 1 2 m product, which is strictly limited to the three-dimensional is Euclidean case with unit pseudoscalar I : sm 3 P (B) (B A) A1 A1 (A B ), (35) A (s) general a b (b a)I (a b)I . (47) degree dependent 3 3 For more on basic geometric algebra I refer to [1,2,9] and to P ( B ) B , P ( B ) B AA1, (36) A 0 0 A n n section 3 of [3]. the exceptions for scalars B and pseudoscalars 0 3. Vector Differential Calculus This section shows how to differentiate functions on linear B being again due to the definition of the inner product in n subspaces of the universal geometric algebra G by vectors. [2], p. 6, (1.21). A projection of one factor of an inner product has the effect It has wide applications particularly to mechanics and physics in general [1]. Separate concepts of gradient, divergence and a P(b) P(a) P(b) P(a) b. (37) curl merge into a single concept of vector derivative, united by the geometric product. For a multivector B G(A ) , with A A we have m m The relationship of differential and derivative is clarified. The Taylor expansion (P. 12) is applied to important examples, (a B) A (a B)A a (BA) if a A 0 . (38) yielding e.g. the Legendre polynomials (P. 36). The Reordering rules for products of homogeneous multivector integrability (P. 42, etc.) of multivector functions are defined are and discussed. Throughout this section a number of basic differentials and derivations are performed explicitly A B (1)r(sr) B A for r s , (39) r s s r illustrating ease and power of the calculus. ([1]-[5]). As for the notation: P. 7 refers to proposition 7 of this A B (1)rs B A . (40) r s s r section. Def. 13 refers to definition 13 of this section. (6) Elementary combinations that occur often are refers to equation number (6) in the previous section on basic geometric algebra. a (b c) (a b)c (a c)b a bc a cb, (41) Standard definitions of continuity and scalar differentiability apply to multivector-valued functions, F F((x)), (x) scalar valued function.
Load more

Recommended publications
  • Eigenvalues and Eigenvectors
    Jim Lambers MAT 605 Fall Semester 2015-16 Lecture 14 and 15 Notes These notes correspond to Sections 4.4 and 4.5 in the text. Eigenvalues and Eigenvectors In order to compute the matrix exponential eAt for a given matrix A, it is helpful to know the eigenvalues and eigenvectors of A. Definitions and Properties Let A be an n × n matrix. A nonzero vector x is called an eigenvector of A if there exists a scalar λ such that Ax = λx: The scalar λ is called an eigenvalue of A, and we say that x is an eigenvector of A corresponding to λ. We see that an eigenvector of A is a vector for which matrix-vector multiplication with A is equivalent to scalar multiplication by λ. Because x is nonzero, it follows that if x is an eigenvector of A, then the matrix A − λI is singular, where λ is the corresponding eigenvalue. Therefore, λ satisfies the equation det(A − λI) = 0: The expression det(A−λI) is a polynomial of degree n in λ, and therefore is called the characteristic polynomial of A (eigenvalues are sometimes called characteristic values). It follows from the fact that the eigenvalues of A are the roots of the characteristic polynomial that A has n eigenvalues, which can repeat, and can also be complex, even if A is real. However, if A is real, any complex eigenvalues must occur in complex-conjugate pairs. The set of eigenvalues of A is called the spectrum of A, and denoted by λ(A). This terminology explains why the magnitude of the largest eigenvalues is called the spectral radius of A.
    [Show full text]
  • Algebra of Linear Transformations and Matrices Math 130 Linear Algebra
    Then the two compositions are 0 −1 1 0 0 1 BA = = 1 0 0 −1 1 0 Algebra of linear transformations and 1 0 0 −1 0 −1 AB = = matrices 0 −1 1 0 −1 0 Math 130 Linear Algebra D Joyce, Fall 2013 The products aren't the same. You can perform these on physical objects. Take We've looked at the operations of addition and a book. First rotate it 90◦ then flip it over. Start scalar multiplication on linear transformations and again but flip first then rotate 90◦. The book ends used them to define addition and scalar multipli- up in different orientations. cation on matrices. For a given basis β on V and another basis γ on W , we have an isomorphism Matrix multiplication is associative. Al- γ ' φβ : Hom(V; W ) ! Mm×n of vector spaces which though it's not commutative, it is associative. assigns to a linear transformation T : V ! W its That's because it corresponds to composition of γ standard matrix [T ]β. functions, and that's associative. Given any three We also have matrix multiplication which corre- functions f, g, and h, we'll show (f ◦ g) ◦ h = sponds to composition of linear transformations. If f ◦ (g ◦ h) by showing the two sides have the same A is the standard matrix for a transformation S, values for all x. and B is the standard matrix for a transformation T , then we defined multiplication of matrices so ((f ◦ g) ◦ h)(x) = (f ◦ g)(h(x)) = f(g(h(x))) that the product AB is be the standard matrix for S ◦ T .
    [Show full text]
  • Math 217: Multilinearity of Determinants Professor Karen Smith (C)2015 UM Math Dept Licensed Under a Creative Commons By-NC-SA 4.0 International License
    Math 217: Multilinearity of Determinants Professor Karen Smith (c)2015 UM Math Dept licensed under a Creative Commons By-NC-SA 4.0 International License. A. Let V −!T V be a linear transformation where V has dimension n. 1. What is meant by the determinant of T ? Why is this well-defined? Solution note: The determinant of T is the determinant of the B-matrix of T , for any basis B of V . Since all B-matrices of T are similar, and similar matrices have the same determinant, this is well-defined—it doesn't depend on which basis we pick. 2. Define the rank of T . Solution note: The rank of T is the dimension of the image. 3. Explain why T is an isomorphism if and only if det T is not zero. Solution note: T is an isomorphism if and only if [T ]B is invertible (for any choice of basis B), which happens if and only if det T 6= 0. 3 4. Now let V = R and let T be rotation around the axis L (a line through the origin) by an 21 0 0 3 3 angle θ. Find a basis for R in which the matrix of ρ is 40 cosθ −sinθ5 : Use this to 0 sinθ cosθ compute the determinant of T . Is T othogonal? Solution note: Let v be any vector spanning L and let u1; u2 be an orthonormal basis ? for V = L . Rotation fixes ~v, which means the B-matrix in the basis (v; u1; u2) has 213 first column 405.
    [Show full text]
  • MATH 304 Linear Algebra Lecture 24: Scalar Product. Vectors: Geometric Approach
    MATH 304 Linear Algebra Lecture 24: Scalar product. Vectors: geometric approach B A B′ A′ A vector is represented by a directed segment. • Directed segment is drawn as an arrow. • Different arrows represent the same vector if • they are of the same length and direction. Vectors: geometric approach v B A v B′ A′ −→AB denotes the vector represented by the arrow with tip at B and tail at A. −→AA is called the zero vector and denoted 0. Vectors: geometric approach v B − A v B′ A′ If v = −→AB then −→BA is called the negative vector of v and denoted v. − Vector addition Given vectors a and b, their sum a + b is defined by the rule −→AB + −→BC = −→AC. That is, choose points A, B, C so that −→AB = a and −→BC = b. Then a + b = −→AC. B b a C a b + B′ b A a C ′ a + b A′ The difference of the two vectors is defined as a b = a + ( b). − − b a b − a Properties of vector addition: a + b = b + a (commutative law) (a + b) + c = a + (b + c) (associative law) a + 0 = 0 + a = a a + ( a) = ( a) + a = 0 − − Let −→AB = a. Then a + 0 = −→AB + −→BB = −→AB = a, a + ( a) = −→AB + −→BA = −→AA = 0. − Let −→AB = a, −→BC = b, and −→CD = c. Then (a + b) + c = (−→AB + −→BC) + −→CD = −→AC + −→CD = −→AD, a + (b + c) = −→AB + (−→BC + −→CD) = −→AB + −→BD = −→AD. Parallelogram rule Let −→AB = a, −→BC = b, −−→AB′ = b, and −−→B′C ′ = a. Then a + b = −→AC, b + a = −−→AC ′.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Matrix Multiplication. Diagonal Matrices. Inverse Matrix. Matrices
    MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. Matrices Definition. An m-by-n matrix is a rectangular array of numbers that has m rows and n columns: a11 a12 ... a1n a21 a22 ... a2n . .. . am1 am2 ... amn Notation: A = (aij )1≤i≤n, 1≤j≤m or simply A = (aij ) if the dimensions are known. Matrix algebra: linear operations Addition: two matrices of the same dimensions can be added by adding their corresponding entries. Scalar multiplication: to multiply a matrix A by a scalar r, one multiplies each entry of A by r. Zero matrix O: all entries are zeros. Negative: −A is defined as (−1)A. Subtraction: A − B is defined as A + (−B). As far as the linear operations are concerned, the m×n matrices can be regarded as mn-dimensional vectors. Properties of linear operations (A + B) + C = A + (B + C) A + B = B + A A + O = O + A = A A + (−A) = (−A) + A = O r(sA) = (rs)A r(A + B) = rA + rB (r + s)A = rA + sA 1A = A 0A = O Dot product Definition. The dot product of n-dimensional vectors x = (x1, x2,..., xn) and y = (y1, y2,..., yn) is a scalar n x · y = x1y1 + x2y2 + ··· + xnyn = xk yk . Xk=1 The dot product is also called the scalar product. Matrix multiplication The product of matrices A and B is defined if the number of columns in A matches the number of rows in B. Definition. Let A = (aik ) be an m×n matrix and B = (bkj ) be an n×p matrix.
    [Show full text]
  • Clifford Algebra with Mathematica
    Clifford Algebra with Mathematica J.L. ARAGON´ G. ARAGON-CAMARASA Universidad Nacional Aut´onoma de M´exico University of Glasgow Centro de F´ısica Aplicada School of Computing Science y Tecnolog´ıa Avanzada Sir Alwyn William Building, Apartado Postal 1-1010, 76000 Quer´etaro Glasgow, G12 8QQ Scotland MEXICO UNITED KINGDOM [email protected] [email protected] G. ARAGON-GONZ´ ALEZ´ M.A. RODRIGUEZ-ANDRADE´ Universidad Aut´onoma Metropolitana Instituto Polit´ecnico Nacional Unidad Azcapotzalco Departamento de Matem´aticas, ESFM San Pablo 180, Colonia Reynosa-Tamaulipas, UP Adolfo L´opez Mateos, 02200 D.F. M´exico Edificio 9. 07300 D.F. M´exico MEXICO MEXICO [email protected] [email protected] Abstract: The Clifford algebra of a n-dimensional Euclidean vector space provides a general language comprising vectors, complex numbers, quaternions, Grassman algebra, Pauli and Dirac matrices. In this work, we present an introduction to the main ideas of Clifford algebra, with the main goal to develop a package for Clifford algebra calculations for the computer algebra program Mathematica.∗ The Clifford algebra package is thus a powerful tool since it allows the manipulation of all Clifford mathematical objects. The package also provides a visualization tool for elements of Clifford Algebra in the 3-dimensional space. clifford.m is available from https://github.com/jlaragonvera/Geometric-Algebra Key–Words: Clifford Algebras, Geometric Algebra, Mathematica Software. 1 Introduction Mathematica, resulting in a package for doing Clif- ford algebra computations. There exists some other The importance of Clifford algebra was recognized packages and specialized programs for doing Clif- for the first time in quantum field theory.
    [Show full text]
  • Linear Algebra Chapter 5: Norms, Inner Products and Orthogonality
    MATH 532: Linear Algebra Chapter 5: Norms, Inner Products and Orthogonality Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Spring 2015 [email protected] MATH 532 1 Outline 1 Vector Norms 2 Matrix Norms 3 Inner Product Spaces 4 Orthogonal Vectors 5 Gram–Schmidt Orthogonalization & QR Factorization 6 Unitary and Orthogonal Matrices 7 Orthogonal Reduction 8 Complementary Subspaces 9 Orthogonal Decomposition 10 Singular Value Decomposition 11 Orthogonal Projections [email protected] MATH 532 2 Vector Norms Vector[0] Norms 1 Vector Norms 2 Matrix Norms Definition 3 Inner Product Spaces Let x; y 2 Rn (Cn). Then 4 Orthogonal Vectors n T X 5 Gram–Schmidt Orthogonalization & QRx Factorizationy = xi yi 2 R i=1 6 Unitary and Orthogonal Matrices n X ∗ = ¯ 2 7 Orthogonal Reduction x y xi yi C i=1 8 Complementary Subspaces is called the standard inner product for Rn (Cn). 9 Orthogonal Decomposition 10 Singular Value Decomposition 11 Orthogonal Projections [email protected] MATH 532 4 Vector Norms Definition Let V be a vector space. A function k · k : V! R≥0 is called a norm provided for any x; y 2 V and α 2 R 1 kxk ≥ 0 and kxk = 0 if and only if x = 0, 2 kαxk = jαj kxk, 3 kx + yk ≤ kxk + kyk. Remark The inequality in (3) is known as the triangle inequality. [email protected] MATH 532 5 Vector Norms Remark Any inner product h·; ·i induces a norm via (more later) p kxk = hx; xi: We will show that the standard inner product induces the Euclidean norm (cf.
    [Show full text]
  • Eigenvalues and Eigenvectors
    Section 5.1 Eigenvalues and Eigenvectors At this point in the class, we understand that the operation of matrix multiplication is very different from scalar multiplication. Matrix multiplication is an operation that finds the product of a pair of matrices, whereas scalar multiplication finds the product of a scalar with a matrix. More to the point, scalar multiplication of a scalar k with a matrix X is an easy operation to perform; indeed there's a simple algorithm: multiply each entry of X by the scalar k. Matrix multiplication on the other hand cannot be expressed using a simple algorithm. Indeed, the operation is extremely complex: to multiply a pair of n×n matrices, it takes about n2 operations, a hugely inefficient process. Thus it is surprising that, for any square matrix A, certain matrix multiplications \collapse" (in a sense) into scalar multiplication. As an example, consider the matrices ( ) ( ) ( ) 3 2 −4 10 A = ; w = ; x = : 3 4 −1 15 The product Aw is not particularly interesting: ( ) −14 Aw = ; −16 indeed, there appears to be very little connection between A, w, and Aw. On the other hand, an we see an interesting phenomenon when we calculate Ax: ( ) ( ) 60 10 Ax = = 6 = 6x: 90 15 Forgetting the calculation in the middle, we have Ax = 6x; in a sense, the matrix multiplication Ax collapsed into scalar multiplication 6x. We will see below that the phenomenon described here is more common than one might think. Identifying products like Ax = 6x will allow us to extract a great deal of useful information about A; accordingly, the rest of the section will be devoted to this phenomenon, which we give a name in the next definition.
    [Show full text]
  • Lecture 12: Abstract Vector Spaces and Concrete Examples
    LECTURE 12 Abstract Vector Spaces and Concrete Examples Our discussion of linear algebra so far has been devoted to discussing the relations between systems of linear equations, matrices, and vectors.As such, topics like the notions of subspaces, bases, linear independence, etc.tend to come off as a bunch of formal nonsense.To see the real utility of these ideas, it’s actually helpful to first abstract things further; this will allow us to bring more interesting examples can be drawn into the picture. Definition 12.1. A vector space over R is a set V for which the following operations are defined • scalar multiplication: for every r ∈ R and v ∈ V we have a map (r, v) → rv ∈ V. • vector addition: for every pair of vectors u, v ∈ V we have a map (u, v) → u + v ∈ V and satisfy the following properties 1.( u + v)+w = u +(v + w) (associativity of vector addition) 2. u + v = v + u (commutativity of vector addition) 3. There exists an element 0 ∈ V such that v + 0 = v for all v ∈ V . (additive identity.) 4. For each v ∈ V there exists an element −v ∈ V such that v +(−v)=0. (additive inverses) 5. r (u + v)=ru + rv (scalar multiplication is distributive with respect to vector addition). 6.( r + s) v = rv + sv. (scalar multiplication is distributive with respect to addition of scalars) 7. r(sv)=(rs)v (scalar multiplication preserves associativity of multiplication in R.) 8.(1) v = v (preservation of scale). Definition 12.2. A subspace W of a vector space V is a subset of V that is closed under scalar multipli- cation and vector addition.
    [Show full text]
  • MATLAB Linear Algebra Tutorial
    Introduction to Linear Algebra using MATLAB Tutorial on Material Covered in ENG EK 127 Relevant to Linear Algebra By Stormy Attaway Reference: Stormy Attaway, MATLAB: A Practical Introduction to Programming and Problem Solving, pp.452+x, Burlington, MA, Elsevier Inc., 2009. MATLAB Basics Windows and Prompt Variables and Assignment Statements Constants Operators and Precedence Built-in functions, Help Types Vectors and Matrices in MATLAB Creating Vector and Matrix Variables . Row Vectors . Column Vectors . Matrices Special Matrix Functions Referring to Elements Dimensions Vectors and Matrices as Function Arguments Matrix Definitions Matrix Operations . Array Operations (term-by-term) . Matrix Multiplication . Inverse . Matrix Augmentation Vector Operations: Dot Product and Cross Product Introduction to Linear Algebra Systems of Equations . Matrix Form 2 x 2 Systems . Elementary Row Operations . Gauss Elimination . Gauss-Jordan Elimination . Reduced Row Echelon Form (RREF) RREF to solve Ax=b for x RREF to find inverse The solve Function in the Symbolic Math Toolbox © Copyright August 2010 by Stormy Attaway Page 1 MATLAB Basics ............................................................................................................ 3 Windows and Prompt.................................................................................................. 3 Variables and Assignment Statements........................................................................ 3 Constants....................................................................................................................
    [Show full text]