Linear Algebra: Vectors


 Michael Kennedy
 5 years ago
 Views:
Transcription
1 A Linear Algebra: Vectors A
2 Appendix A: LINEAR ALGEBRA: VECTORS TABLE OF CONTENTS Page A Motivation A 3 A2 Vectors A 3 A2 Notational Conventions A 4 A22 Visualization A 5 A23 Special Vectors A 5 A3 Vector Operations A 5 A3 Transposition A 6 A32 Equality A 6 A33 Addition and Subtraction A 6 A34 Multiplication and Division by Scalar A 7 A35 Span A 7 A36 Inner Product, Norm and Length A 7 A37 Unit Vectors and Normalization A 9 A38 Angles and Orthonormality A 9 A39 Orthogonal Projection A 0 A30 Orthogonal Bases and Subspaces A 0 A Exercises A 2 A Solutions to Exercises A 3 A 2
3 A2 VECTORS A Motivation Matrix notation was invented primarily to express linear algebra relations in compact form Compactness enhances visualization and understanding of essentials To illustrate this point, consider the following set of m linear relations between one set of n quantities, x,, x n, and another set of m quantities, y, y 2,, y m : a x + a a j x j + + a n x n = y a 2 + a a 2 j x j + + a 2n x n = y 2 a i x + a i2 + + a ij x j + + a in x n = y i (A) a m x + a m2 + + a mj x j + + a mn x n = y m The subscripted a, x and y quantities that appear in this set of relations may be formally arranged as follows: a a 2 a j a n x y a 2 a 22 a 2 j a 2n y 2 a i a i2 a ij a in x j = (A2) y i a m a m2 a mj a mn x n y m Two kinds of mathematical objects can be distinguished in (A2) The twodimensional array expression enclosed in brackets is a matrix, which we call A Matrices are defined and studied in Appendix B The onedimensional array expressions in brackets are column vectors or simply vectors, which we call x and y, respectively The use of boldface uppercase and lowercase letters to denote matrices and vectors, respectively, follows conventional matrixnotation rules in engineering applications See A2 for conevntions used in this book Replacing the expressions in (A2) by these symbols we obtain the matrix form of (A): Ax= y (A3) Putting A next to x means matrix product of A times x, which is a generalization of the ordinary scalar multiplication, and follows the rules explained later Clearly (A3) is a more compact, short hand form of (A) Another key practical advantage of the matrix notation is that it translates directly to the computer implementation of linear algebra processes in languages that offer array data structures Largely by Arthur Cayley (82 895) at Cambridge (UK) in 858 [30] with substantial contributions from his colleague James Joseph Sylvester (84 897) See Appendix H for additional historical details related to the developmnent of FEM Curiously, vectors for mathematical physics (that is, living in 3D space) were independently created by J Willard Gibbs ( ) and Oliver Heaviside ( ) in the late XIX century [72] The overlap between matrices and vectors in n dimensions was not established until the XX century A 3
4 Appendix A: LINEAR ALGEBRA: VECTORS A2 Vectors We begin by defining a vector, a set of n numbers which we shall write in the form x = x x n (A4) This object is called a column vector We shall see later that although a vector may be viewed as a special case of a matrix, it deserves treatment on its own The symbol x is the name of the vector If n numbers are arranged in a horizontal array, as in z = [ z z 2 z n ], (A5) then z is called a row vector If the term vector is used without a qualifier, it is understood to be a column vector such as (A4) A2 Notational Conventions Typeset vectors will be designated by bold lowercase letters For example: a, b, c, x, y, z (A6) On the other hand, handwritten or typewritten vectors, which are those written on paper or on the blackboard, are identified by putting a wiggle or bar underneath the letter For example: ã (A7) Subscripted quantities such as x in (A4) are called the entries or components 2 of x, while n is called the order of the vector x Vectors of order one (n = ) are called scalars These are the usual quantities of analysis For compactness one sometimes abbreviates the phrase vector of order n to just nvector For example, z in equation (A8) below is a 4vector If the components are real numbers, the vector is called a real vector If the components are complex numbers we have a complex vector Linear algebra embraces complex vectors as easily as it does real ones; however, we rarely need complex vectors for most of our exposition Consequently real vectors will be assumed unless otherwise noted 2 The term component is primarily used in mathematical treatments whereas entry is used more in conjunction with the computer implementation The term element is also used in the literature but this will be avoided here (as well as in companion books) as it may lead to confusion with finite elements A 4
5 (a) x = [ x (b) x x 3 ] x = x 3 A3 VECTOR OPERATIONS x x Figure A Standard visualization of 2vectors and 3vectors as position vectors in 2space and 3space, respectively Example A 4 2 x = 3 0 is real column vector of order 4, or, briefly, a 4vector (A8) Example A2 is a real row vector of order 6 q = [ ] (A9) Occasionally we shall use the shorthand component notation x = [x i ], (A0) for a generic vector This comes handy when it is desirable to show the notational scheme obeyed by components A22 Visualization To help visualization, a twodimensional vector x (n = 2) can be depicted as a line segment, or arrow, directed from the chosen origin to a point on the Cartesian plane of the paper with coordinates (x, ) See Figure A(a) In mechanics this is called a position vector in 2space One may resort to a similar geometrical interpretation in threedimensional Cartesian space (n = 3) See Figure A(b) Drawing becomes a bit messier, however, although some knowledge of perspective and projective geometry helps The interpretation extends to Euclidean spaces of dimensions n > 3 but direct visualization is of course impaired A23 Special Vectors The null vector, written 0, is the vector all of whose components are zero The unit vector, denoted by e i, is the vector all of whose components are zero, except the i th component, which is one After introducing matrices (Appendix B) a unit vector may be defined as the i th column of the identity matrix The unitary vector, called e, is the vector all of whose components are unity A 5
6 Appendix A: LINEAR ALGEBRA: VECTORS A3 Vector Operations Operations on vectors in twodimensional and threedimensional space are extensively studied in courses on Mathematical Physics Here we summarize operations on ncomponent vectors that are most useful from the standpoint of the development of finite elements A3 Transposition The transpose of a column vector x is the row vector that has the same components, and is denoted by x T : x T = [ x x n ] (A) Similarly, the transpose of a row vector is the column vector that has the same components Transposing a vector twice yields the original vector: (x T ) T = x Example A3 The transpose of [ ] 3 a = 6 and transposing b gives back a is b = a T = [ 3 6] A32 Equality Two column vectors x and y of equal order n are said to be equal if and only if their components are equal, x i = y i, for all i =,n We then write x = y Similarly for row vectors Two vectors of different order cannot be compared for equality or inequality A row vector cannot be directly compared to a column vector of the same order (unless their dimension is ); one of the two has to be transposed before a comparison can be made A33 Addition and Subtraction The simplest operation acting on two vectors is addition The sum of two vectors of same order n, x and y, is written x + y and defined to be the vector of order n x + y def = x + y + y 2 x n + y n (A2) If x and y are not of the same order, the addition operation is undefined The operation is commutative: x + y = y + x, and associative: x + (y + z) = (x + y) + z Strictly speaking, the plus sign connecting x and y is not the same as the sign connecting x i and y i However, since it enjoys the same analytical properties, there is no harm in using the same symbol in both cases The geometric interpretation of the vector addition operator for two and threedimensional vectors (n = 2, 3) is the well known paralellogram law See Figure A2 For n = the usual scalar addition results For vector subtraction, replace + by in the foregoing expressions A 6
7 A3 VECTOR OPERATIONS y = [ y y 2 ] [ ] x + y x + y = + y 2 x = [ x ] Figure A2 In two and threedimensional space, the vector addition operation is equivalent to the well known parallelogram composition law Example A4 The sum of [ ] 3 a = 6 and b = [ 2 4 ] is a + b = [ ] A34 Multiplication and Division by Scalar Multiplication of a vector x by a scalar c is defined by means of the relation c x def = cx cx 2 cx n (A3) This operation is often called scaling of a vector The geometrical interpretation of scaling is: the scaled vector points the same way, but its magnitude is multiplied by c If c = 0, the result is the null vector If c < 0 the direction of the vector is reversed In paticular, if c = the resulting operation ( )x = x is called reflexion about the origin or simply reflexion Division of a vector by a scalar c 0 is equivalent to multiplication by /c The operation is written The operation is not defined if c = 0 x c x/c def = (/c)x (A4) A35 Span Sometimes it is not just x which is of interest but the line determined by x, namely the collection cx of all scalar multiples of x, including x We call this Span(x) Note that the span always includes the null vector The span of two vectors, x and y with common origin, is the set of vectors c x + c 2 y for arbitrary scaling coefficients c and c 2 If the two vectors are not parallel, the geometric visualization of the span is that of the plane defined by the two vectors Hence the statement: a plane is spanned by two noncoincident vectors, responds to this interpretation The span of an arbitrary number of vectors of common origin is studied in A30 A 7
8 Appendix A: LINEAR ALGEBRA: VECTORS A36 Inner Product, Norm and Length The inner product of two column vectors x and y, of same order n, is a scalar function denoted by (x, y) as well as two other forms shown below and is defined by (x, y) = x T y = y T x def = n i= x i y i sc = x i y i (A5) This operation is also called dot product and interior product If the two vectors are not of the same order, the inner product is undefined The two other notations shown in (A5), namely x T y and y T x, exhibit the inner product as a special case of the matrix product discussed in Appendix B The last expression in (A5) applies the socalled Einstein s summation convention, which implies sum on repeated indices (in this case, i ) and allows the operand to be dropped The inner product is commutative: (x, y) = (y, x) but not generally associative: (x,(y, z)) ((x, y), z) If n = it reduces to the usual scalar product The scalar product is of course associative Example A5 The inner product of ] a = [ 3 6 and b = [ 2 4 ] is (a, b) = a T b = ( ) + 6 ( 4) = 9 Remark A The inner product is not the only way of multiplying two vectors There are other vector products, such as the cross or outer product, which are important in many applications such as fluid mechanics and nonlinear dynamics They are not treated here because they are not defined when going from vectors to matrices Furthermore they are not easily generalized for vectors of order beyond 3 The Euclidean norm or 2norm of a real vector x is a scalar denoted by x that results by taking the inner product of the vector with itself: x def = (x, x) = n i= i sc = x i x i (A6) Because the norm is a sum of squares, it is zero only if x is the null vector It thus provides a meter (mathematically, a norm) on the vector magnitude The Euclidean length or simply length of a real vector, denoted by x, is the positive square root of its Euclidean norm: x def = + x (A7) This definition agrees with the intuitive concept of vector magnitude in two and threedimensional space (n = 2, 3) For n = observe that the length of a scalar x is its absolute value x, hence the notation x The Euclidean norm is not the only vector norm used in practice, but it will be sufficient for the present book A 8
9 A3 VECTOR OPERATIONS Two important inequalities satisfied by vector norms (and not just the Euclidean norm) are the CauchySchwarz inequality: (x, y) x y, (A8) and the triangle inequality: x + y x + y (A9) Example A6 The Euclidean norm of [ ] 4 x = 3 (A20) is x = = 25, and its length is x = 25 = 5 A37 Unit Vectors and Normalization A vector of length one is called a unit vector Any nonnull vector x can be scaled to unit length by dividing all components by its original length: x / x x x/ x = 2 / x x n / x (A2) This particular scaling is called normalization to unit length If x is the null vector, the normalization operation is undefined Example A7 The unitvector normalization of [ ] 4 x = is x/ x =x/5 = 3 [ ] (A22) A38 Angles and Orthonormality The angle in radians between two unit real vectors x and y, written (x, y), is the real number θ satisfying 0 θ π (or 0 θ 80 if measured in degrees) The value is defined by the cosine formula: cos θ = (x, y) (A23) If the vectors are not of unit length, they should be normalized to such, and so the general formula for two arbitrary vectors is cos θ = ( x x, y (x, y) ) = y x y (A24) The angle θ formed by a nonnull vector with itself is zero This will always be the case for n = If one of the vectors is null, the angle is undefined The definition (A24) agrees with that of the angle formed by oriented lines in two and threedimensional Euclidean geometry (n = 2, 3) The generalization to n dimensions is a natural one A 9
10 Appendix A: LINEAR ALGEBRA: VECTORS For some applications the acute angle φ between the line on x and the line on y (ie, between the vector spans) is more appropriate than θ This angle between Span(x) and Span(y) is the real number φ satisfying 0 φ π/2 and cos φ = Two vectors x and y connected by the relation (x, y) x y (A25) (x, y) = 0, (A26) are said to be orthogonal The acute angle formed by two orthogonal vectors is π/2 radians or 90 A39 Orthogonal Projection The orthogonal projection of a vector y onto a vector x is the vector p that has the direction of x and is orthogonal to y p The condition can be stated as (p, y p) = 0, in which p = c x x (A27) It is easily shown that c = (y, x ) = y cos θ, x (A28) where θ is the angle between x and y Ifx and y are orthogonal, the projection of any of them onto the other vanishes A30 Orthogonal Bases and Subspaces Let b k, k =,m be a set of ndimensional unit vectors which are mutually orthogonal Then if a particular ndimensional vector x admits the representation x = m c k b k k= (A29) the coefficients c k are given by the inner products c k = (x, b k ) = n i= x i b k i sc = x i b k i (A30) We also have Parseval s equality (x, x) = m k= c 2 k sc = c k c k (A3) If the representation (A29) holds, the set b k is called an orthonormal basis for the vector x The b k are called the base vectors A 0
11 A3 VECTOR OPERATIONS The set of all vectors x given by (A30) forms a subspace of dimension m The subspace is said to be spanned by the basis b k, and is called Span (b k ) 3 The numbers c k are called the coordinates of x with respect to that basis If m = n, the set b k forms a complete orthonormal basis for the ndimensional space (The qualifier complete means that all ndimensional vectors are representable in terms of such a basis) The simplest complete orthonormal basis is of order n is the ndimensional Cartesian basis b = 0 0, b2 = 0 0, bn = 0 0 In this case the position coordinates of x are simply its components, that is, c k x k (A32) 3 Note that if m = we have the span of a single vector, which as noted in A35 is simply a line passing through the origin of coordinates A
12 Appendix A: LINEAR ALGEBRA: VECTORS Homework Exercises for Appendix A: Vectors EXERCISE A Given the fourdimensional vectors x = 4, y = (EA) (a) compute the Euclidean norms and lengths of x and y; (b) compute the inner product (x,y); (c) verify that the inequalities (A8) (A9) hold; (d) normalize x and y to unit length; (e) compute the angles θ = (x, y) and φ given by (?) and (?) (f) compute the projection of y onto x and verify that the orthogonality condition (A28) holds EXERCISE A2 Given the base vectors b = 7 0, b 2 = (EA2) (a) (b) Check that b and b 2 are orthonormal, ie, orthogonal and of unit length Compute the coefficients c and c 2 in the following representation: 8 6 z = 2 = c b + c 2 b 2 26 (EA3) (c) Using the values computed in (b), verify that the coordinateexpansion formulas and Parseval s equality are correct EXERCISE A3 Prove that Parseval s equality holds in general EXERCISE A4 What are the angles θ and φ formed by an arbitrary nonnull vector x and the opposite vector x? EXERCISE A5 Show that (αx,βy) = αβ (x, y) where x and y are arbitrary vectors, and α and β are scalars (EA4) A 2
13 Solutions to Exercises Homework Exercises for Appendix A  Solutions EXERCISE A (a) (b) (c) (d) (e) (f) EXERCISE A2 (a) x = ( 8) 2 = 00, x =0 y = 2 + ( 5) = 00, y =0 (x, y) = = 30 (x, y) =30 x y =00 x + y = ( ) 2 + ( ) 2 + ( 3) 2 = 40 x + y =0 + 0 = 20 x u = x 0 = , y u = y 0 = cos θ = (x u, y u ) = 030, θ = 0746 cos φ = (x u, y u ) =030, φ = 7254 c = y cos θ = p = c x u 04 2 = 3 04 = y p = (p, y p) = = 0 b =0 2 + ( 07) =, b = b 2 = ( 05) =, b 2 = (b, b 2 ) = = 0 (b) c = 30, c 2 = 0 (by inspection, or solving a system of 2 linear equations) (c) c = (z, b ) = = 30 c 2 = (z, b 2 ) = = 0 c 2 + c2 2 = = 000 = z =8 2 + ( 6) 2 + ( 2) = 000 A 3
14 Appendix A: LINEAR ALGEBRA: VECTORS EXERCISE A3 See any linear algebra book, eg [699] EXERCISE A4 EXERCISE A5 (summation convention used) θ = 80, φ = 0 (αx,βy) = αx i βy i = αβ x i y i = αβ (x, y) A 4
Linear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More informationLecture L3  Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3  Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More information1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two nonzero vectors u and v,
1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It
More informationUnified Lecture # 4 Vectors
Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More information13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.
3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in threespace, we write a vector in terms
More information9.4. The Scalar Product. Introduction. Prerequisites. Learning Style. Learning Outcomes
The Scalar Product 9.4 Introduction There are two kinds of multiplication involving vectors. The first is known as the scalar product or dot product. This is socalled because when the scalar product of
More informationFigure 1.1 Vector A and Vector F
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
More information28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. v x. u y v z u z v y u y u z. v y v z
28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.4 Cross Product 1.4.1 Definitions The cross product is the second multiplication operation between vectors we will study. The goal behind the definition
More informationLinear Algebra Review. Vectors
Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length
More informationv w is orthogonal to both v and w. the three vectors v, w and v w form a righthanded set of vectors.
3. Cross product Definition 3.1. Let v and w be two vectors in R 3. The cross product of v and w, denoted v w, is the vector defined as follows: the length of v w is the area of the parallelogram with
More informationThe Dot and Cross Products
The Dot and Cross Products Two common operations involving vectors are the dot product and the cross product. Let two vectors =,, and =,, be given. The Dot Product The dot product of and is written and
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationVector and Matrix Norms
Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a nonempty
More informationSection 9.1 Vectors in Two Dimensions
Section 9.1 Vectors in Two Dimensions Geometric Description of Vectors A vector in the plane is a line segment with an assigned direction. We sketch a vector as shown in the first Figure below with an
More informationRecall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.
ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the ndimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?
More informationGeometry of Vectors. 1 Cartesian Coordinates. Carlo Tomasi
Geometry of Vectors Carlo Tomasi This note explores the geometric meaning of norm, inner product, orthogonality, and projection for vectors. For vectors in threedimensional space, we also examine the
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More information2. Spin Chemistry and the Vector Model
2. Spin Chemistry and the Vector Model The story of magnetic resonance spectroscopy and intersystem crossing is essentially a choreography of the twisting motion which causes reorientation or rephasing
More informationMechanics 1: Vectors
Mechanics 1: Vectors roadly speaking, mechanical systems will be described by a combination of scalar and vector quantities. scalar is just a (real) number. For example, mass or weight is characterized
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationF Matrix Calculus F 1
F Matrix Calculus F 1 Appendix F: MATRIX CALCULUS TABLE OF CONTENTS Page F1 Introduction F 3 F2 The Derivatives of Vector Functions F 3 F21 Derivative of Vector with Respect to Vector F 3 F22 Derivative
More informationVectors Math 122 Calculus III D Joyce, Fall 2012
Vectors Math 122 Calculus III D Joyce, Fall 2012 Vectors in the plane R 2. A vector v can be interpreted as an arro in the plane R 2 ith a certain length and a certain direction. The same vector can be
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationMATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.
MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationA linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form
Section 1.3 Matrix Products A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form (scalar #1)(quantity #1) + (scalar #2)(quantity #2) +...
More information5.3 The Cross Product in R 3
53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or
More information9 Multiplication of Vectors: The Scalar or Dot Product
Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation
More informationSection V.3: Dot Product
Section V.3: Dot Product Introduction So far we have looked at operations on a single vector. There are a number of ways to combine two vectors. Vector addition and subtraction will not be covered here,
More informationLecture 2 Matrix Operations
Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrixvector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or
More informationExcel supplement: Chapter 7 Matrix and vector algebra
Excel supplement: Chapter 7 atrix and vector algebra any models in economics lead to large systems of linear equations. These problems are particularly suited for computers. The main purpose of this chapter
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationdiscuss how to describe points, lines and planes in 3 space.
Chapter 2 3 Space: lines and planes In this chapter we discuss how to describe points, lines and planes in 3 space. introduce the language of vectors. discuss various matters concerning the relative position
More informationA vector is a directed line segment used to represent a vector quantity.
Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationRotation Matrices and Homogeneous Transformations
Rotation Matrices and Homogeneous Transformations A coordinate frame in an ndimensional space is defined by n mutually orthogonal unit vectors. In particular, for a twodimensional (2D) space, i.e., n
More informationThe Matrix Elements of a 3 3 Orthogonal Matrix Revisited
Physics 116A Winter 2011 The Matrix Elements of a 3 3 Orthogonal Matrix Revisited 1. Introduction In a class handout entitled, ThreeDimensional Proper and Improper Rotation Matrices, I provided a derivation
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationOrthogonal Projections
Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationChapter 17. Orthogonal Matrices and Symmetries of Space
Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length
More informationMath 215 HW #6 Solutions
Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationAdding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors
1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number
More information6. Vectors. 1 20092016 Scott Surgent (surgent@asu.edu)
6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,
More informationISOMETRIES OF R n KEITH CONRAD
ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x
More informationDot product and vector projections (Sect. 12.3) There are two main ways to introduce the dot product
Dot product and vector projections (Sect. 12.3) Two definitions for the dot product. Geometric definition of dot product. Orthogonal vectors. Dot product and orthogonal projections. Properties of the dot
More informationv 1 v 3 u v = (( 1)4 (3)2, [1(4) ( 2)2], 1(3) ( 2)( 1)) = ( 10, 8, 1) (d) u (v w) = (u w)v (u v)w (Relationship between dot and cross product)
0.1 Cross Product The dot product of two vectors is a scalar, a number in R. Next we will define the cross product of two vectors in 3space. This time the outcome will be a vector in 3space. Definition
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationOne advantage of this algebraic approach is that we can write down
. Vectors and the dot product A vector v in R 3 is an arrow. It has a direction and a length (aka the magnitude), but the position is not important. Given a coordinate axis, where the xaxis points out
More informationSection 6.1  Inner Products and Norms
Section 6.1  Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,
More information5. Orthogonal matrices
L Vandenberghe EE133A (Spring 2016) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 51 Orthonormal
More informationIntroduction to Matrix Algebra
Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra  1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary
More information1 Vectors: Geometric Approach
c F. Waleffe, 2008/09/01 Vectors These are compact lecture notes for Math 321 at UWMadison. Read them carefully, ideally before the lecture, and complete with your own class notes and pictures. Skipping
More informationAP Physics  Vector Algrebra Tutorial
AP Physics  Vector Algrebra Tutorial Thomas Jefferson High School for Science and Technology AP Physics Team Summer 2013 1 CONTENTS CONTENTS Contents 1 Scalars and Vectors 3 2 Rectangular and Polar Form
More informationVector Spaces; the Space R n
Vector Spaces; the Space R n Vector Spaces A vector space (over the real numbers) is a set V of mathematical entities, called vectors, U, V, W, etc, in which an addition operation + is defined and in which
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationSolving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
More informationCross product and determinants (Sect. 12.4) Two main ways to introduce the cross product
Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product Geometrical definition Properties Expression in components. Definition in components Properties Geometrical expression.
More informationSection 4.4 Inner Product Spaces
Section 4.4 Inner Product Spaces In our discussion of vector spaces the specific nature of F as a field, other than the fact that it is a field, has played virtually no role. In this section we no longer
More informationSection 5.3. Section 5.3. u m ] l jj. = l jj u j + + l mj u m. v j = [ u 1 u j. l mj
Section 5. l j v j = [ u u j u m ] l jj = l jj u j + + l mj u m. l mj Section 5. 5.. Not orthogonal, the column vectors fail to be perpendicular to each other. 5..2 his matrix is orthogonal. Check that
More informationx1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.
Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability
More informationMethods for Finding Bases
Methods for Finding Bases Bases for the subspaces of a matrix Rowreduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,
More informationVector Algebra II: Scalar and Vector Products
Chapter 2 Vector Algebra II: Scalar and Vector Products We saw in the previous chapter how vector quantities may be added and subtracted. In this chapter we consider the products of vectors and define
More informationT ( a i x i ) = a i T (x i ).
Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)
More informationOctober 3rd, 2012. Linear Algebra & Properties of the Covariance Matrix
Linear Algebra & Properties of the Covariance Matrix October 3rd, 2012 Estimation of r and C Let rn 1, rn, t..., rn T be the historical return rates on the n th asset. rn 1 rṇ 2 r n =. r T n n = 1, 2,...,
More information1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
More informationMATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.
MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all ndimensional column
More information18.06 Problem Set 4 Solution Due Wednesday, 11 March 2009 at 4 pm in 2106. Total: 175 points.
806 Problem Set 4 Solution Due Wednesday, March 2009 at 4 pm in 206 Total: 75 points Problem : A is an m n matrix of rank r Suppose there are righthandsides b for which A x = b has no solution (a) What
More informationPhysics 235 Chapter 1. Chapter 1 Matrices, Vectors, and Vector Calculus
Chapter 1 Matrices, Vectors, and Vector Calculus In this chapter, we will focus on the mathematical tools required for the course. The main concepts that will be covered are: Coordinate transformations
More informationNotes on Orthogonal and Symmetric Matrices MENU, Winter 2013
Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,
More informationFURTHER VECTORS (MEI)
Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level  MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: 97 Mathematics
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationReview A: Vector Analysis
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Review A: Vector Analysis A... A0 A.1 Vectors A2 A.1.1 Introduction A2 A.1.2 Properties of a Vector A2 A.1.3 Application of Vectors
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 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
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More informationLectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain
Lectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n realvalued matrix A is said to be an orthogonal
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationGeometric Transformations
Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted
More information1 Symmetries of regular polyhedra
1230, notes 5 1 Symmetries of regular polyhedra Symmetry groups Recall: Group axioms: Suppose that (G, ) is a group and a, b, c are elements of G. Then (i) a b G (ii) (a b) c = a (b c) (iii) There is an
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More informationVector has a magnitude and a direction. Scalar has a magnitude
Vector has a magnitude and a direction Scalar has a magnitude Vector has a magnitude and a direction Scalar has a magnitude a brick on a table Vector has a magnitude and a direction Scalar has a magnitude
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More informationTypical Linear Equation Set and Corresponding Matrices
EWE: Engineering With Excel Larsen Page 1 4. Matrix Operations in Excel. Matrix Manipulations: Vectors, Matrices, and Arrays. How Excel Handles Matrix Math. Basic Matrix Operations. Solving Systems of
More informationCS3220 Lecture Notes: QR factorization and orthogonal transformations
CS3220 Lecture Notes: QR factorization and orthogonal transformations Steve Marschner Cornell University 11 March 2009 In this lecture I ll talk about orthogonal matrices and their properties, discuss
More informationGeometric Algebra Primer. Jaap Suter
Geometric Algebra Primer Jaap Suter March 12, 2003 Abstract Adopted with great enthusiasm in physics, geometric algebra slowly emerges in computational science. Its elegance and ease of use is unparalleled.
More informationLinear Algebra Done Wrong. Sergei Treil. Department of Mathematics, Brown University
Linear Algebra Done Wrong Sergei Treil Department of Mathematics, Brown University Copyright c Sergei Treil, 2004, 2009, 2011, 2014 Preface The title of the book sounds a bit mysterious. Why should anyone
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationEquations, Inequalities & Partial Fractions
Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities
More informationVector Spaces. Chapter 2. 2.1 R 2 through R n
Chapter 2 Vector Spaces One of my favorite dictionaries (the one from Oxford) defines a vector as A quantity having direction as well as magnitude, denoted by a line drawn from its original to its final
More informationVector Math Computer Graphics Scott D. Anderson
Vector Math Computer Graphics Scott D. Anderson 1 Dot Product The notation v w means the dot product or scalar product or inner product of two vectors, v and w. In abstract mathematics, we can talk about
More information3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or
More informationComputing Orthonormal Sets in 2D, 3D, and 4D
Computing Orthonormal Sets in 2D, 3D, and 4D David Eberly Geometric Tools, LLC http://www.geometrictools.com/ Copyright c 19982016. All Rights Reserved. Created: March 22, 2010 Last Modified: August 11,
More information