sin(α + θ) = sin α cos θ + cos α sin θ cos(α + θ) = cos α cos θ sin α sin θ
|
|
- Stuart Gregory
- 5 years ago
- Views:
Transcription
1 Rotations in the 2D Plane Trigonometric addition formulas: sin(α + θ) = sin α cos θ + cos α sin θ cos(α + θ) = cos α cos θ sin α sin θ Rotate coordinates by angle θ: 1. Start with x = r cos α y = r sin α 2. Rotated coordinates are x = r cos(α + θ) =r(cos α cos θ sin α sin θ) y = r sin(α + θ) =r(sin α cos θ + cos α sin θ) 3. Substituting gives x = x cos θ y sin θ y = x sin θ + y cos θ Matrix version of rotated coordinates: [ ] [ x cos θ sin θ y = sin θ cos θ ][ ] x y 1
2 Complex Numbers Both multiplication and addition are associative and commutative. If x = a + bi, define the conjugate of x, denoted x, tobea bi. Define the norm of x, denoted x, by x 2 = x x = (a + bi) (a bi) = a 2 + b 2 xy = x y Let x=a+bi, then x x x 2 = 1 and therefore x 1 = a a 2 + b 2 b a 2 + b 2 i Complex numbers form a field. They also form a normed division algebra. Multiplication by a complex number with norm one is a rotation. 2
3 Rotations in 3D Space cos θ sin θ 0 z-axis: R z = sin θ cos θ x-axis: R x = cos θ sin θ 0 sin θ cos θ sin θ 0 cos θ y-axis: R y = cos θ 0 sin θ 3
4 Rotation about an Arbitrary Axis 1. Rotate around the z-axis until the arbitrary axis is in the zx-plane. 2. Rotate around the y-axis until the arbitrary axis coincides with the x-axis. 3. Rotate around the x-axis by angle θ. 4. Undo the rotations around the y-axis and z-axis. A = R T z RT y R xr y R z If the unit vector (n x,n y,n z ) is in the direction of the arbitrary axis, then rotation around this axis corresponds to multiplying by the following matrix where t =(1 cos θ): n 2 xt + cos θ n x n y t n x sin θ n x n z t + n y sin θ A = n x n y t + n x sin θ n 2 yt + cos θ n y n z t n x sin θ n x n z t n y sin θ n y n z t + n x sin θ n 2 z t + cos θ 4
5 Some Linear Algebra Rotation is a linear transformation and therefore can be represented by a matrix. Rotation preserves distances and angles. Therefore, it preserves dot products. Av Aw = v w If v = 1 0 and w = , then Av Aw = a 11 a 12 + a 21 a 22 + a 31 a 32 = =0 All the columns of A are perpendicular to each other. Av Av = a 11 a 11 + a 21 a 21 + a 31 a 31 = 1. All columns have unit length. A is an orthonormal matrix. A T A = I (det A) 2 =1 Since a rotation preserves handedness of the coordinate system, det A = 1 A matrix is a rotation matrix if and only if it is orthonormal with determinant 1. Proposition: Composition of two rotations is a rotation. Proof: Product of orthonormal matrices with determinant 1 is orthonormal with determinant 1. 5
6 Finding the Axis and Angle of a Rotation Matrix Axis: Since A is a rotation, we know that the matrix equation Ax = x has a solution. Solving gives (a 23 a 32, a 31 a 13, a 12 a 21 ) Angle of rotation: tr(a) =tr((cr x )C T ) = tr(c T CR x ) = tr(r x ) = cos θ. [ ] tr(a) 1 θ = cos 1 2 6
7 Quaternion Histroy Sir William Rowan Hamilton Lived in Dublin Discovered Quaternions on October 16, 1843 file:///c /Documents%20and%20Settings/sjanke/Desktop/Quaternion/QHistory1.html10/11/2005 8:35:08 AM
8 Quaternion Histroy Plaque on Brougham Bridge over the Royal Canal Commemorating Hamilton's discovery of quaternions on October 16, 1843 file:///c /Documents%20and%20Settings/sjanke/Desktop/Quaternion/QHistory2.html10/11/2005 8:35:22 AM
9 Quaternion Algebra: Take all objects of the form a + bi + cj + dk. Define ij = k, jk = i, ki = j, i 2 = 1, j 2 = 1, k 2 = 1 Note ij = ji, jk = kj, ik = ki. Use the distributive property to multiply: (a 1 + b 1 i + c 1 j + d 1 k) (a 2 + b 2 i + c 2 j + d 2 k)=(a 1 a 2 b 1 b 2 c 1 c 2 d 1 d 2 ) +(a 1 b 2 + a 2 b 1 + c 1 d 2 c 2 d 1 )i +(a 1 c 2 + a 2 c 1 + b 2 d 1 b 1 d 2 )j +(a 1 d 2 + a 2 d 1 + b 1 c 2 b 2 c 1 )k qr does not necessarily equal rq. Quaternion algebra is not commutative. 7
10 More Quaternion Algebra Define the conjugate of q by q = a bi cj dk. Then q q = a 2 + b 2 + c 2 + d 2 and define q = q q. We have qp = q p. The quaternions form a skew field and is a normed division algebra. There is a natural correspondence between the number q = a + bi + cj + dk and the vector (a, b, c, d). Think of a quaternion as a scalar plus a 3D vector: q = a + bi + cj + dk = a + v = q (cos α + sin α n) Think of a quaternion as a combination of complex numbers C 1 = a 1 + b 1 i and C 2 = a 2 + b 2 i. q = C 1 + C 2 j = a 1 + b 1 i + a 2 j + b 2 k 8
11 Yet More Quaternion Algebra Take quaternions p = p 0 + p and q = q 0 + q, then pq =(p 0 q 0 p q )+p 0 q + q 0 p + p q Take a quaternion q and a vector v in R 3. q v q =(q 2 0 q 2 ) v +2q 0 ( q v )+2( q v ) q Suppose v = k q, then q v q =(q0 2 q 2 )k q +2q 0 ( q k q)+2( q k q) q =(q0 2 q 2 )k q +2( q k q) q = q0k q 2 k q 2 q +2k q 2 q = k(q0 2 + q 2 ) q = k q Rotation: If q = cos θ 2 +sin θ 2 q, then q v q is the vector resulting from rotating v around axis q by angle θ. Multiplying two unit quaternions composes the two rotations: q 2 (q 1 v q 1 ) q 2 =(q 2 q 1 ) v ( q 1 q 2 ) = q v q 9
12 Conversion of Quaternion to Matrix Start with a unit quaternion q = d + ai + bj + ck The the corresponding rotation matrix is: 1 2b2 2c 2 2ab +2dc 2ac 2db 2ab 2dc 1 2a 2 2c 2 2bc +2da 2ac +2db 2bc 2da 1 2a 2 2b 2 10
13 Operation Counts Applying Matrix: 9 multiplications and 6 additions Applying Quaternion: 23 multiplications and 8 additions Compose Matrices: 27 multiplications and 18 additions Compose Quaternions: 12 multiplications and 32 additions Convert Quaternion to Matrix: 16 multiplications and 10 additions Storage: Matrix (9 floating point), Quaternion(4 floating point) 11
14 Calculating Quaternion Multiplication q 1 = d 1 + a 1 i + b 1 j + c 1 k q 2 = d 2 + a 2 i + b 2 j + c 2 k q = d + ai + bj + ck q = q 1 q 2 A =(d 1 + a 1 )(d 2 + a 2 ) B =(c 1 b 1 )(b 2 c 2 ) C =(a 1 d 1 )(b 2 c 2 ) D =(b 1 + c 1 )(a 2 d 2 ) E =(a 1 + c 1 )(a 2 + b 2 ) F =(a 1 c 1 )(a 2 b 2 ) G =(d 1 + b 1 )(d 2 c 2 ) H =(d 1 b 1 )(d 2 + c 2 ) d = B +( E F + G + H)/2 a = A (E + F + G + H)/2 b = C +(E F + G H)/2 c = D +(E F G + H)/2 8 Multiplications 4 Divisions 32 Additions 12
15 Spherical Interpolation of Quaternions slerp(q 1, q 2,t)= sin((1 t)φ) q 1 + sin(αt) sin φ sin φ q 2 13
16 Quaternion Histroy History of Vector Analysis Hamilton discovers quaternions 1843 Gauss may have discovered quaternions independently somewhat earlier. Mobius: Barycentric coordinates Grassmann 1844 Tait and Maxwell (1873 treatise using quaternions and standard vectors) Gibbs and Heaviside 1880's Debate in Nature 1890's (Tait: "Don't spoon feed the public.") file:///c /Documents%20and%20Settings/sjanke/Desktop/Quaternion/QHistory3.html10/11/2005 8:35:37 AM
17 The Normed Division Algebras R (real numbers) is a commutative associative normed division algebra (with trivial conjugation). C (complex numbers) is a commutative associative normed division algebra with (nontrivial conjugation). H (quaternions) is a associative normed division algebra (non-commutative). O (octonions) is a normed division algebra (non-associative and non-commutative). Lemma: Z=Y+iY is a division algebra just when Y is an associative division algebra. 14
Course 2BA1: Hilary Term 2007 Section 8: Quaternions and Rotations
Course BA1: Hilary Term 007 Section 8: Quaternions and Rotations David R. Wilkins Copyright c David R. Wilkins 005 Contents 8 Quaternions and Rotations 1 8.1 Quaternions............................ 1 8.
More informationECS 178 Course Notes QUATERNIONS
ECS 178 Course Notes QUATERNIONS Kenneth I. Joy Institute for Data Analysis and Visualization Department of Computer Science University of California, Davis Overview The quaternion number system was discovered
More informationQuaternions. Basilio Bona. Semester 1, DAUIN Politecnico di Torino. B. Bona (DAUIN) Quaternions Semester 1, / 40
Quaternions Basilio Bona DAUIN Politecnico di Torino Semester 1, 2016-2017 B. Bona (DAUIN) Quaternions Semester 1, 2016-2017 1 / 40 Introduction Complex numbers with unit norm can be used as rotation operators
More informationComplex Numbers and Quaternions for Calc III
Complex Numbers and Quaternions for Calc III Taylor Dupuy September, 009 Contents 1 Introduction 1 Two Ways of Looking at Complex Numbers 1 3 Geometry of Complex Numbers 4 Quaternions 5 4.1 Connection
More informationA (Mostly) Linear Algebraic Introduction to Quaternions
A (Mostly) Linear Algebraic Introduction to Quaternions Joe McMahon Program in Applied Mathematics University of Arizona Fall 23 1 Some History 1.1 Hamilton s Discovery and Subsequent Vandalism Having
More informationThe Quaternions & Octonions: A Basic introduction to Their Algebras. By: Kyle McAllister. Boise State University
The Quaternions & Octonions: A Basic introduction to Their Algebras By: Kyle McAllister Boise State University McAllister The Quaternions and Octonions have a storied history, one with a restless supporter,
More informationHypercomplex numbers
Hypercomplex numbers Johanna Rämö Queen Mary, University of London j.ramo@qmul.ac.uk We have gradually expanded the set of numbers we use: first from finger counting to the whole set of positive integers,
More informationSubalgebras of the Split Octonions
Subalgebras of the Split Octonions Lida Bentz July 27, 2017 CONTENTS CONTENTS Contents 1 Introduction 3 1.1 Complex numbers......................... 3 1.2 Quaternions............................ 3 1.3
More informationCourse MA2C02, Hilary Term 2010 Section 4: Vectors and Quaternions
Course MA2C02, Hilary Term 2010 Section 4: Vectors and Quaternions David R. Wilkins Copyright c David R. Wilkins 2000 2010 Contents 4 Vectors and Quaternions 47 4.1 Vectors...............................
More informationContents. D. R. Wilkins. Copyright c David R. Wilkins
MA232A Euclidean and Non-Euclidean Geometry School of Mathematics, Trinity College Michaelmas Term 2017 Section 5: Vector Algebra and Spherical Trigonometry D. R. Wilkins Copyright c David R. Wilkins 2015
More informationQuaternions 2 AUI Course Denbigh Starkey
Quaternions 2 AUI Course Denbigh Starkey 1. Background 2 2. Some Basic Quaternion Math 4 3. The Justification of the Quaternion Rotation Formula 5 4. Interpolation between two Unit Quaternions SLERP vs.
More informationComputing Moore-Penrose Inverses of Ore Polynomial Matrices Yang Zhang
Computing Moore-Penrose Inverses of Ore Polynomial Matrices Yang Zhang Department of Mathematics University of Manitoba, Canada Outline History and motivation. Theorems and algorithms for quaternion polynomials
More informationLecture 7. Quaternions
Matthew T. Mason Mechanics of Manipulation Spring 2012 Today s outline Motivation Motivation have nice geometrical interpretation. have advantages in representing rotation. are cool. Even if you don t
More informationQUATERNIONS AND ROTATIONS
QUATERNIONS AND ROTATIONS SVANTE JANSON 1. Introduction The purpose of this note is to show some well-known relations between quaternions and the Lie groups SO(3) and SO(4) (rotations in R 3 and R 4 )
More informationThe Quaternions. The Quaternions. John Huerta. Department of Mathematics UC Riverside. Cal State Stanislaus
John Huerta Department of Mathematics UC Riverside Cal State Stanislaus The Complex Numbers The complex numbers C form a plane. Their operations are very related to two dimensional geometry. In particular,
More information5. Vector Algebra and Spherical Trigonometry (continued)
MA232A Euclidean and Non-Euclidean Geometry School of Mathematics, Trinity College Michaelmas Term 2017 Section 5: Vector Algebra and Spherical Trigonometry David R. Wilkins 5.1. Vectors in Three-Dimensional
More informationUnderstanding Quaternions: Rotations, Reflections, and Perspective Projections. Ron Goldman Department of Computer Science Rice University
Understanding Quaternions: Rotations, Reflections, and Perspective Projections Ron Goldman Department of Computer Science Rice University The invention of the calculus of quaternions is a step towards
More informationQuaternions and the four-square theorem
Quaternions and the four-square theorem Background on Hamilton s Quaternions We have the complex numbers C = {a + b i : a, b R} an their integral analogue, [i] = {a + bi : a, b R}, the Gaussian integers,
More informationEXERCISES. a b = a + b l aq b = ab - (a + b) + 2. a b = a + b + 1 n0i) = oii + ii + fi. A. Examples of Rings. C. Ring of 2 x 2 Matrices
/ rings definitions and elementary properties 171 EXERCISES A. Examples of Rings In each of the following, a set A with operations of addition and multiplication is given. Prove that A satisfies all the
More information12.1. Cartesian Space
12.1. Cartesian Space In most of your previous math classes, we worked with functions on the xy-plane only meaning we were working only in 2D. Now we will be working in space, or rather 3D. Now we will
More informationMA232A Euclidean and Non-Euclidean Geometry School of Mathematics, Trinity College Michaelmas Term 2017 Vector Algebra and Spherical Trigonometry
MA232A Euclidean and Non-Euclidean Geometry School of Mathematics, Trinity College Michaelmas Term 2017 Vector Algebra and Spherical Trigonometry David R. Wilkins 3.1. Vectors in Three-Dimensional Euclidean
More informationChapter XI Novanion rings
Chapter XI Novanion rings 11.1 Introduction. In this chapter we continue to provide general structures for theories in physics. J. F. Adams proved in 1960 that the only possible division algebras are at
More informationElementary maths for GMT
Elementary maths for GMT Linear Algebra Part 2: Matrices, Elimination and Determinant m n matrices The system of m linear equations in n variables x 1, x 2,, x n a 11 x 1 + a 12 x 2 + + a 1n x n = b 1
More informationQuaternions and their applications
Quaternions and their applications Ramachandran Subramanian February 20, 2014 Jack B. Kuipers. Quaternions and Rotation Sequences. university press, New Jersey, 1998 Princeton Outline 1 Background 2 Introduction
More informationMODEL ANSWERS TO HWK #7. 1. Suppose that F is a field and that a and b are in F. Suppose that. Thus a = 0. It follows that F is an integral domain.
MODEL ANSWERS TO HWK #7 1. Suppose that F is a field and that a and b are in F. Suppose that a b = 0, and that b 0. Let c be the inverse of b. Multiplying the equation above by c on the left, we get 0
More informationLinear Algebra: Matrix Eigenvalue Problems
CHAPTER8 Linear Algebra: Matrix Eigenvalue Problems Chapter 8 p1 A matrix eigenvalue problem considers the vector equation (1) Ax = λx. 8.0 Linear Algebra: Matrix Eigenvalue Problems Here A is a given
More informationBe fruitful and multiply Genesis 1:28
Lecture 15: Quaternions: Multiplication in the Space of Mass-Points Be fruitful and multiply Genesis 1:8 15.1 Vector Spaces and Division Algebras The space of mass-points is a 4-dimensional vector space:
More informationMODEL ANSWERS TO THE FIRST HOMEWORK
MODEL ANSWERS TO THE FIRST HOMEWORK 1. Chapter 4, 1: 2. Suppose that F is a field and that a and b are in F. Suppose that a b = 0, and that b 0. Let c be the inverse of b. Multiplying the equation above
More informationLie Algebra of Unit Tangent Bundle in Minkowski 3-Space
INTERNATIONAL ELECTRONIC JOURNAL OF GEOMETRY VOLUME 12 NO. 1 PAGE 1 (2019) Lie Algebra of Unit Tangent Bundle in Minkowski 3-Space Murat Bekar (Communicated by Levent Kula ) ABSTRACT In this paper, a one-to-one
More informationWe could express the left side as a sum of vectors and obtain the Vector Form of a Linear System: a 12 a x n. a m2
Week 22 Equations, Matrices and Transformations Coefficient Matrix and Vector Forms of a Linear System Suppose we have a system of m linear equations in n unknowns a 11 x 1 + a 12 x 2 + + a 1n x n b 1
More informationQuaternions and Octonions
Quaternions and Octonions Alberto Elduque Universidad de Zaragoza 1 Real and complex numbers 2 Quaternions 3 Rotations in three-dimensional space 4 Octonions 2 / 32 1 Real and complex numbers 2 Quaternions
More informationWhy Transforms? Want to animate objects and camera Translations Rotations Shears And more.. Want to be able to use projection transforms
Why Transforms? Want to animate objects and camera Translations Rotations Shears And more.. Want to be able to use projection transforms ITCS 3050:Game Engine Programming 1 Geometric Transformations Implementing
More informationUnderstanding Quaternions. Ron Goldman Department of Computer Science Rice University
Understanding Quaternions Ron Goldman Department of Computer Science Rice University The invention of the calculus of quaternions is a step towards the knowledge of quantities related to space which can
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F ) denotes the collection of all m n matrices over F Matrices will usually be denoted
More informationQuaternions. Mike Bailey. Computer Graphics Quaternions.pptx
1 Quaternions This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License Mike Bailey mjb@cs.oregonstate.edu Quaternions.pptx A Useful Concept: Spherical
More informationQuaternions and Groups
Quaternions and Groups Rich Schwartz October 16, 2014 1 What is a Quaternion? A quaternion is a symbol of the form a+bi+cj +dk, where a,b,c,d are real numbers and i,j,k are special symbols that obey the
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K R MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND First Printing, 99 Chapter LINEAR EQUATIONS Introduction to linear equations A linear equation in n unknowns x,
More informationMath 3108: Linear Algebra
Math 3108: Linear Algebra Instructor: Jason Murphy Department of Mathematics and Statistics Missouri University of Science and Technology 1 / 323 Contents. Chapter 1. Slides 3 70 Chapter 2. Slides 71 118
More informationChapter 7. Linear Algebra: Matrices, Vectors,
Chapter 7. Linear Algebra: Matrices, Vectors, Determinants. Linear Systems Linear algebra includes the theory and application of linear systems of equations, linear transformations, and eigenvalue problems.
More informationa 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 11 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,, a n, b are given real
More informationQuaternions An Algebraic View (Supplement)
Quaternions Algebraic View (Supplement) 1 Quaternions An Algebraic View (Supplement) Note. You likely first encounter the quaternions in Introduction to Modern Algebra. John Fraleigh s A First Course in
More information1 Matrices and Systems of Linear Equations
March 3, 203 6-6. Systems of Linear Equations Matrices and Systems of Linear Equations An m n matrix is an array A = a ij of the form a a n a 2 a 2n... a m a mn where each a ij is a real or complex number.
More informationLinear Equations and Matrix
1/60 Chia-Ping Chen Professor Department of Computer Science and Engineering National Sun Yat-sen University Linear Algebra Gaussian Elimination 2/60 Alpha Go Linear algebra begins with a system of linear
More informationQuaternion Algebras. Edgar Elliott. May 1, 2016
Quaternion Algebras Edgar Elliott May 1, 2016 Copyright (C) 2016 Edgar Elliott. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License,
More informationA Tutorial on Euler Angles and Quaternions
A Tutorial on Euler Angles and Quaternions Moti Ben-Ari Department of Science Teaching Weimann Institute of Science http://www.weimann.ac.il/sci-tea/benari/ Version.0.1 c 01 17 b Moti Ben-Ari. This work
More informationChapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form
Chapter 5. Linear Algebra A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are real numbers. 1
More informationDUAL SPLIT QUATERNIONS AND SCREW MOTION IN MINKOWSKI 3-SPACE * L. KULA AND Y. YAYLI **
Iranian Journal of Science & echnology, ransaction A, Vol, No A Printed in he Islamic Republic of Iran, 6 Shiraz University DUAL SPLI UAERNIONS AND SCREW MOION IN MINKOWSKI -SPACE L KULA AND Y YAYLI Ankara
More informationA matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More informationA matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More information7 Matrix Operations. 7.0 Matrix Multiplication + 3 = 3 = 4
7 Matrix Operations Copyright 017, Gregory G. Smith 9 October 017 The product of two matrices is a sophisticated operations with a wide range of applications. In this chapter, we defined this binary operation,
More informationEquality: Two matrices A and B are equal, i.e., A = B if A and B have the same order and the entries of A and B are the same.
Introduction Matrix Operations Matrix: An m n matrix A is an m-by-n array of scalars from a field (for example real numbers) of the form a a a n a a a n A a m a m a mn The order (or size) of A is m n (read
More informationLinear Algebra. Paul Yiu. 6D: 2-planes in R 4. Department of Mathematics Florida Atlantic University. Fall 2011
Linear Algebra Paul Yiu Department of Mathematics Florida Atlantic University Fall 2011 6D: 2-planes in R 4 The angle between a vector and a plane The angle between a vector v R n and a subspace V is the
More informationChapter 6 Additional Topics in Trigonometry, Part II
Chapter 6 Additional Topics in Trigonometry, Part II Section 3 Section 4 Section 5 Vectors in the Plane Vectors and Dot Products Trigonometric Form of a Complex Number Vocabulary Directed line segment
More informationGeometric Algebra. Gary Snethen Crystal Dynamics
Geometric Algebra Gary Snethen Crystal Dynamics gsnethen@crystald.com Questions How are dot and cross products related? Why do cross products only exist in 3D? Generalize cross products to any dimension?
More informationDS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.
DS-GA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1
More informationGeometric Fundamentals in Robotics Quaternions
Geometric Fundamentals in Robotics Quaternions Basilio Bona DAUIN-Politecnico di Torino July 2009 Basilio Bona (DAUIN-Politecnico di Torino) Quaternions July 2009 1 / 47 Introduction Quaternions were discovered
More informationSystems of Linear Equations and Matrices
Chapter 1 Systems of Linear Equations and Matrices System of linear algebraic equations and their solution constitute one of the major topics studied in the course known as linear algebra. In the first
More informationAssignment 10. Arfken Show that Stirling s formula is an asymptotic expansion. The remainder term is. B 2n 2n(2n 1) x1 2n.
Assignment Arfken 5.. Show that Stirling s formula is an asymptotic expansion. The remainder term is R N (x nn+ for some N. The condition for an asymptotic series, lim x xn R N lim x nn+ B n n(n x n B
More informationIntroduction to quaternions
. Introduction Introduction to uaternions Invented and developed by William Hamilton in 843, uaternions are essentially a generalization of complex numbers to four dimensions (one real dimension, three
More informationBe fruitful and multiply Genesis 1:28
Chapter 15: Quaternions: Multiplication in the Space of Mass-Points Be fruitful and multiply Genesis 1:8 1. Vector Spaces and Division Algebras The space of mass-points is a 4-dimensional vector space:
More informationSystems of Linear Equations and Matrices
Chapter 1 Systems of Linear Equations and Matrices System of linear algebraic equations and their solution constitute one of the major topics studied in the course known as linear algebra. In the first
More informationInverses of Square Matrices
Inverses of Square Matrices A. Havens Department of Mathematics University of Massachusetts, Amherst February 23-26, 2018 Outline 1 Defining Inverses Inverses for Products and Functions Defining Inverse
More informationCS 246 Review of Linear Algebra 01/17/19
1 Linear algebra In this section we will discuss vectors and matrices. We denote the (i, j)th entry of a matrix A as A ij, and the ith entry of a vector as v i. 1.1 Vectors and vector operations A vector
More informationGroup Theory: Math30038, Sheet 6
Group Theory: Math30038, Sheet 6 Solutions GCS 1. Consider the group D ofrigidsymmetriesofaregularn-gon (which may be turned over). Prove that this group has order 2n, is non-abelian, can be generated
More informationMATH 106 LINEAR ALGEBRA LECTURE NOTES
MATH 6 LINEAR ALGEBRA LECTURE NOTES FALL - These Lecture Notes are not in a final form being still subject of improvement Contents Systems of linear equations and matrices 5 Introduction to systems of
More informationANoteontheRepresentationandDefinitionofDualSplitSemiQuaternionsAlgebra
Global Journal of Science Frontier Research: F Mathematics Decision Sciences Volume 7 Iue 8 Version.0 Year 07 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals
More informationReview of linear algebra
Review of linear algebra 1 Vectors and matrices We will just touch very briefly on certain aspects of linear algebra, most of which should be familiar. Recall that we deal with vectors, i.e. elements of
More informationAbstract Algebra II Groups ( )
Abstract Algebra II Groups ( ) Melchior Grützmann / melchiorgfreehostingcom/algebra October 15, 2012 Outline Group homomorphisms Free groups, free products, and presentations Free products ( ) Definition
More informationSUMS OF SQUARES. Lecture at Vanier College, Sept. 23, Eyal Goren, McGill University.
SUMS OF SQUARES Lecture at Vanier College, Sept. 23, 2011. Eyal Goren, McGill University. email: eyal.goren@mcgill.ca Two Squares. Which numbers are a sum of two squares? More precisely, which positive
More informationA = 3 B = A 1 1 matrix is the same as a number or scalar, 3 = [3].
Appendix : A Very Brief Linear ALgebra Review Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics Very often in this course we study the shapes
More informationQuaternions. R. J. Renka 11/09/2015. Department of Computer Science & Engineering University of North Texas. R. J.
Quaternions R. J. Renka Department of Computer Science & Engineering University of North Texas 11/09/2015 Definition A quaternion is an element of R 4 with three operations: addition, scalar multiplication,
More informationMatrix Algebra & Elementary Matrices
Matrix lgebra & Elementary Matrices To add two matrices, they must have identical dimensions. To multiply them the number of columns of the first must equal the number of rows of the second. The laws below
More informationSPLIT QUATERNIONS and CANAL SURFACES. in MINKOWSKI 3-SPACE
INTERNATIONAL JOURNAL OF GEOMETRY Vol. 5 (016, No., 51-61 SPLIT QUATERNIONS and CANAL SURFACES in MINKOWSKI 3-SPACE SELAHATTIN ASLAN and YUSUF YAYLI Abstract. A canal surface is the envelope of a one-parameter
More informationUnit Generalized Quaternions in Spatial Kinematics
Vol (07) - Pages 7-4 Unit Generalized Quaternions in Spatial Kinematics Mehdi Jafari, echnical Vocational University, Urmia, Iran, mj_msc@yahoo.com Abstract- After a review of some fundamental properties
More informationVectors Summary. can slide along the line of action. not restricted, defined by magnitude & direction but can be anywhere.
Vectors Summary A vector includes magnitude (size) and direction. Academic Skills Advice Types of vectors: Line vector: Free vector: Position vector: Unit vector (n ): can slide along the line of action.
More informationA VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 2010
A VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 00 Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics
More informationLinear Algebra March 16, 2019
Linear Algebra March 16, 2019 2 Contents 0.1 Notation................................ 4 1 Systems of linear equations, and matrices 5 1.1 Systems of linear equations..................... 5 1.2 Augmented
More information= 1 and 2 1. T =, and so det A b d
Chapter 8 Determinants The founder of the theory of determinants is usually taken to be Gottfried Wilhelm Leibniz (1646 1716, who also shares the credit for inventing calculus with Sir Isaac Newton (1643
More informationLinear Algebra. Workbook
Linear Algebra Workbook Paul Yiu Department of Mathematics Florida Atlantic University Last Update: November 21 Student: Fall 2011 Checklist Name: A B C D E F F G H I J 1 2 3 4 5 6 7 8 9 10 xxx xxx xxx
More information12.5 Equations of Lines and Planes
12.5 Equations of Lines and Planes Equation of Lines Vector Equation of Lines Parametric Equation of Lines Symmetric Equation of Lines Relation Between Two Lines Equations of Planes Vector Equation of
More informationRings If R is a commutative ring, a zero divisor is a nonzero element x such that xy = 0 for some nonzero element y R.
Rings 10-26-2008 A ring is an abelian group R with binary operation + ( addition ), together with a second binary operation ( multiplication ). Multiplication must be associative, and must distribute over
More informationMTH5112 Linear Algebra I MTH5212 Applied Linear Algebra (2017/2018)
MTH5112 Linear Algebra I MTH5212 Applied Linear Algebra (2017/2018) COURSEWORK 3 SOLUTIONS Exercise ( ) 1. (a) Write A = (a ij ) n n and B = (b ij ) n n. Since A and B are diagonal, we have a ij = 0 and
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Second Online Version, December 1998 Comments to the author at krm@maths.uq.edu.au Contents 1 LINEAR EQUATIONS
More informationQuantum Computing Lecture 2. Review of Linear Algebra
Quantum Computing Lecture 2 Review of Linear Algebra Maris Ozols Linear algebra States of a quantum system form a vector space and their transformations are described by linear operators Vector spaces
More informationON A NEW SPECIES OF IMAGINARY QUANTITIES CONNECTED WITH A THEORY OF QUATERNIONS. William Rowan Hamilton
ON A NEW SPECIES OF IMAGINARY QUANTITIES CONNECTED WITH A THEORY OF QUATERNIONS By William Rowan Hamilton (Proceedings of the Royal Irish Academy, 2 (1844), 424 434.) Edited by David R. Wilkins 1999 On
More informationLinear Algebra Review. Vectors
Linear Algebra Review 9/4/7 Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa (UCSD) Cogsci 8F Linear Algebra review Vectors
More informationOctonions? A non-associative geometric algebra. Benjamin Prather. October 19, Florida State University Department of Mathematics
A non-associative geometric algebra Benjamin Florida State University Department of Mathematics October 19, 2017 Let K be a field with 1 1 Let V be a vector space over K. Let, : V V K. Definition V is
More informationMath Problem set # 7
Math 128 - Problem set # 7 Clifford algebras. April 8, 2004 due April 22 Because of disruptions due the the Jewish holidays and surgery on my knee there will be no class next week (April 13 or 15). (Doctor
More informationQuadratics. Shawn Godin. Cairine Wilson S.S Orleans, ON October 14, 2017
Quadratics Shawn Godin Cairine Wilson S.S Orleans, ON Shawn.Godin@ocdsb.ca October 14, 2017 Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 1 / 110 Binary Quadratic Form A form is a homogeneous
More informationICS 6N Computational Linear Algebra Matrix Algebra
ICS 6N Computational Linear Algebra Matrix Algebra Xiaohui Xie University of California, Irvine xhx@uci.edu February 2, 2017 Xiaohui Xie (UCI) ICS 6N February 2, 2017 1 / 24 Matrix Consider an m n matrix
More informationMathematical Fundamentals
Mathematical Fundamentals Ming-Hwa Wang, Ph.D. COEN 148/290 Computer Graphics COEN 166/366 Artificial Intelligence COEN 396 Interactive Multimedia and Game Programming Department of Computer Engineering
More informationGroup, Rings, and Fields Rahul Pandharipande. I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S,
Group, Rings, and Fields Rahul Pandharipande I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S, A binary operation φ is a function, S S = {(x, y) x, y S}. φ
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Corrected Version, 7th April 013 Comments to the author at keithmatt@gmail.com Chapter 1 LINEAR EQUATIONS 1.1
More informationKnowledge Discovery and Data Mining 1 (VO) ( )
Knowledge Discovery and Data Mining 1 (VO) (707.003) Review of Linear Algebra Denis Helic KTI, TU Graz Oct 9, 2014 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, 2014 1 / 74 Big picture: KDDM Probability Theory
More informationApplied Linear Algebra
Applied Linear Algebra Gábor P. Nagy and Viktor Vígh University of Szeged Bolyai Institute Winter 2014 1 / 262 Table of contents I 1 Introduction, review Complex numbers Vectors and matrices Determinants
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K R MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Second Online Version, December 998 Comments to the author at krm@mathsuqeduau All contents copyright c 99 Keith
More informationLinear Algebra: Lecture notes from Kolman and Hill 9th edition.
Linear Algebra: Lecture notes from Kolman and Hill 9th edition Taylan Şengül March 20, 2019 Please let me know of any mistakes in these notes Contents Week 1 1 11 Systems of Linear Equations 1 12 Matrices
More informationClifford Analysis, Homework 1
Clifford Analysis, Homework 1 November 1, 017 1 Let w v 1... v k, for vectors v j. Show that ŵ is the result of negating the vectors: ŵ ( v 1 )... ( v k ). Show that w is the result of changing the order
More informationBSc (Hons) in Computer Games Development. vi Calculate the components a, b and c of a non-zero vector that is orthogonal to
1 APPLIED MATHEMATICS INSTRUCTIONS Full marks will be awarded for the correct solutions to ANY FIVE QUESTIONS. This paper will be marked out of a TOTAL MAXIMUM MARK OF 100. Credit will be given for clearly
More informationMatrix Theory. A.Holst, V.Ufnarovski
Matrix Theory AHolst, VUfnarovski 55 HINTS AND ANSWERS 9 55 Hints and answers There are two different approaches In the first one write A as a block of rows and note that in B = E ij A all rows different
More information