Introduction to Vector Spaces and Subspaces Sections , 4.7
|
|
- Phyllis McDonald
- 7 years ago
- Views:
Transcription
1 Introduction to Vector Spaces and Subspaces Sections , 4.7 Math 81, Applied Analysis Instructor: Dr. Doreen De Leon 1 Why Study Vector Spaces? Vector spaces have many applications. For one, they provide a framework to deal with analytical and geometrical problems, and are used in the Fourier transform. There are also applications of vector spaces in optimization theory. The minimax theorem of game theory stating the existence of a unique payoff when all players play optimally can be formulated and proven using vector space methods. Least squares estimation, which is used in among other areas digital filter design, tracking (Kalman filters), control systems, etc. Representation theory basically transforms problems in abstract algebra, particularly group theory, to problems in linear algebra. Such discussions are, however, beyond the scope of this class. Therefore, we limit ourselves to the above brief discussion, and launch directly into the topic at hand. 2 Introduction to Vector Spaces Sections , Vectors in R n We will use some familiar ideas about vectors to motivate the idea of a vector space, which we will introduce a bit later. First, a vector can be thought of as a directed line segment (or arrow) that has both magnitude and direction. In Calculus III, you probably defined vector addition in two ways: using the parallelogram law, where x + y is the diagonal of the parellelogram formed by x and y; and by adding corresponding elements of x and y to obtain x + y. Then, to form the sum of three vectors, we simply add two vectors as above and then add the third vector to the result. This can be performed equivalently by adding the first two vectors together, and then adding the third vector to the sum; or, by adding the last two 1
2 vectors together, and then adding the first vector to the sum. So, we have that for all vectors x, y, and z x + y = y + x, x + (y + z) = (x + y) + z. The zero vector, denoted 0, is defined as the vector satisfying x + 0 = x, for all vectors x. We consider the zero vector as having zero magnitude and arbitrary direction. Geometrically, we picture the zero vector as corresponding to a point in space. Let x denote the vector that has the same magnitude as x but points in the opposite direction. Then, if we add x and x using either approach above, we obtain x + ( x) = 0. The vector x so defined is the additive inverse of x. fundamental properties of vector addition. The above properties are the We also need to define the operation of multiplication of a vector by a scalar, i.e., scalar multiplication. Geometrically, if x is a vector and c is a scalar, then cx is defined by the vector whose magnitude is c times the magnitude of x and whose direction is the same as x if c > 0 and the opposite of x if c < 0. If c = 0, then cx = 0. Scalar multiplication has several important properties, which can easily be verified, either geometrically or algebraically using the definitions of vector addition and scalar multiplication. For all vectors x and y and all scalars a, and b, 1x = x, (ab)x = a(bx), a(x + y) = ax + ay, (a + b)x = ax + bx. If you notice, we did not define multiplication of vectors. However, from Calculus III, you are familiar with the dot product and cross product of vectors. For the purpose of talking about vector spaces, we will ignore these two operations on vectors, only concerning ourselves with vector addition and scalar multiplication. If we consider vectors in the plane, denoted R 2, then each vector v = (v 1, v 2 ) can be identified with a point in space with coordinates (v 1, v 2 ), and the direction and magnitude of v are determined by considering the arrow with vertex at the origin, (0, 0). Then, for all vectors v = (v 1, v 2 ) and w = (w 1, w 2 ) and all scalars a, we define vector addition and scalar multiplication as v + w = (v 1, v 2 ) + (w 1, w 2 ) = (v 1 + w 1, v 2 + w 2 ), av = a (v 1, v 2 ) = (av 1, av 2 ). We can easily verify the above listed properties for vectors in R 2. For example, 0 = (0, 0) and since (v 1, v 2 ) + ( v 1, v 2 ) = (0, 0), 2
3 the additive inverse of v is v = ( v 1, v 2 ). It is straightforward to extend these ideas to vectors in space, denoted R 3 by simply adding a third component. Each vector v = (v 1, v 2, v 3 ) can be identified with a point in space with coordinates (v 1, v 2, v 3 ), and the direction and magnitude of v are determined by considering the arrow with vertex at the origin, (0, 0, 0). Vector addition and scalar multiplication are defined as follows. If v = (v 1, v 2, v 3 ) and w = (w 1, w 2, w 3 ) and c is a scalar, v + w = (v 1, v 2, v 3 ) + (w 1, w 2, w 3 ) = (v 1 + w 1, v 2 + w 2, v 3 + w 3 ), av = c (v 1, v 2, v 3 ) = (cv 1, cv 2, cv 3 ). And again, we can show that the properties given above for vector addition and scalar multiplication are satisfied by R 3, with 0 = (0, 0, 0) and v = ( v 1, v 2, v 3 ). We can generalize this notion of vectors that we can visualize to vectors with any number of components. The set of all vectors with n real-valued components is denoted by R n. Vector addition and scalar multiplication in R n are defined componentwise. Given any two vectors u and v in R n, defined by u = (u 1, u 2,..., u n ) and v = (v 1, v 2,..., v n ), and any scalar c, the standard vector addition and scalar multiplication are defined as follows: u + v = (u 1 + v 1, u 2 + v 2,..., u n + v n ) cu = (cu 1, cu 2,..., cu n ) We can show that for any positive integer n, R n possesses all of the algebraic properties for vector addition and scalar multiplication as R 2 and R 3. These properties give us the framework to define a general notion, that of vector spaces. 2.2 Vector Spaces We will start with a set of elements V, which we will call vectors. They may be vectors in the sense of vectors discussed previously, or they may be some other mathematical construct, such as functions, matrices, or polynomials. Associated with this set of vectors are two operations, vector addition and scalar multiplication. Vector addition can be thought of as a rule for combining two vectors in V. To simplify things notationally, we will use the + sign to denote vector addition. So, the result of adding two vectors u and v will be denoted u + v. Scalar multiplication can be thought of as a rule for combining each vector in V with any scalar, and we will use the usual notation av to denote the result of multiplying the vector v by the scalar a. So, what is a vector space? 3
4 Definition: Let V be a set of elements called vectors, in which the operations of addition of vectors and multiplication of vectors by scalars are defined. Then, given any vectors u, v, w V and any scalars a and b, V is a vector space if all of the following properties are satisfied. (1) u + v V (V is closed under vector addition). (2) au V (V is closed under scalar multiplication). (3) u + v = v + u (commutativity). (4) u + (v + w) = (u + v) + w (associativity). (5) There exists a zero element in V, 0 V, such that u + 0 = 0 + u = u. (6) There exists an element in V, u, called the additive inverse, such that u + ( u) = u + u = 0. (7) a(u + v) = au + av. (8) (a + b)u = au + bu. (9) a(bu) = (ab)u. (10) (1)u = u. Notation: 4
5 2.3 Subspaces Definition: Theorem 1. Examples: 1) W = {(x, y, z) : z = 0}. Is W a subspace of R 3? Why or why not? 5
6 2) W = {(x, y, z) : y = 1} Is W a subspace of R 3? Why or why not? 3) W = {(x, y, z) : x 2 + y 2 z 2 = 0}. Is W a subspace of R 3? Why or why not? 6
7 2.4 Solution Subspaces Theorem 2. If A is a constant m n matrix, then the solution set of the system is a subspace of R n, called the solution space of the system. Ax = 0 (1) Proof. Let W be the set of all solutions to (1). Verify the conditions for a subspace. 2.5 General Vector Spaces The term vector in vector space can be interpreted in a more general sense. Examples: (1) Given m and n positive integers, define M mn as the set of all m n matrices with real entries. Then M mn is a vector space. Matrices play the role of vectors, with matrix addition defining vector addition multiplication of a matrix by a scalar defining scalar multiplication The zero element is the zero matrix. It is easily verified using these definitions of vector addition and scalar multiplication that M mn satisfies properties (1)-(10) of the definition of a vector space. 7
8 (2) F is the set of all real-valued functions defined on R. Functions play the role of vectors, with vector addition and scalar multiplication defined as follows. For f, g F, (f + g)(x) = f(x) + g(x) vector addition For f F and c R, (cf)(x) = cf(x) scalar multiplication. It is easily verified using these definitions of vector addition and scalar multiplication that F is a vector space, with the zero element being the function f(x) = 0 (i.e., the function whose value is zero for all x). (3) C n [a, b] is the set of all continuous functions with n continuous derivatives on [a, b]. C n [a, b] is a vector space with the same properties as F. (4) P is the set of all polynomials. P is a vector space with the polynomials playing the role of vectors. Note: P is also a subspace of F since P F. (5) P n is the set of all polynomials of degree at most n. P n is a vector space with the polynomials playing the role of vectors. In fact, P n is a subspace of P. 2.6 Subspaces of General Vector Spaces 8
9 Examples: (1) W is the set of diagonal 2 2 matrices. Is W a subspace of M 22? Why or why not? (2) W is the set of all solutions of the differential equation y + p(x)y = 0. Is W a subspace of C 1 (I)? Why or why not? 9
10 (3) W is the set of all solutions of the differential equation y + p(x)y = x. Is W a subspace of C 1 (I)? Why or why not? (4) W is the set of all polynomials of the form a 0 + a 1 x + a 2 x 2 such that a 0 = 2a 2. Is W a subspace of P 2? Why or why not? 10
11 (5) W is the set of all polynomials in P 2 whose coefficients are odd integers. Is W a subspace of P 2? Why or why not? 11
Linear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional 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 informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationMATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.
MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar
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 informationInner product. Definition of inner product
Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product
More informationMath 4310 Handout - Quotient Vector Spaces
Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable
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 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 informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that
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 informationLinear Algebra I. Ronald van Luijk, 2012
Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.
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 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 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 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 informationLinear Algebra: Vectors
A Linear Algebra: Vectors A 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
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 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 informationThese axioms must hold for all vectors ū, v, and w in V and all scalars c and d.
DEFINITION: A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the following axioms
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 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 information2.3. Finding polynomial functions. An Introduction:
2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned
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 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 informationEssential Mathematics for Computer Graphics fast
John Vince Essential Mathematics for Computer Graphics fast Springer Contents 1. MATHEMATICS 1 Is mathematics difficult? 3 Who should read this book? 4 Aims and objectives of this book 4 Assumptions made
More informationMATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).
MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0
More informationSouth Carolina College- and Career-Ready (SCCCR) Pre-Calculus
South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 3-4 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 informationSystems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
More informationMath 115A - Week 1 Textbook sections: 1.1-1.6 Topics covered: What is a vector? What is a vector space? Span, linear dependence, linear independence
Math 115A - Week 1 Textbook sections: 1.1-1.6 Topics covered: What is Linear algebra? Overview of course What is a vector? What is a vector space? Examples of vector spaces Vector subspaces Span, linear
More informationJUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials
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 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 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 informationMath 1050 Khan Academy Extra Credit Algebra Assignment
Math 1050 Khan Academy Extra Credit Algebra Assignment KhanAcademy.org offers over 2,700 instructional videos, including hundreds of videos teaching algebra concepts, and corresponding problem sets. In
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 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 three-space, we write a vector in terms
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationChapter 7. Matrices. Definition. An m n matrix is an array of numbers set out in m rows and n columns. Examples. ( 1 1 5 2 0 6
Chapter 7 Matrices Definition An m n matrix is an array of numbers set out in m rows and n columns Examples (i ( 1 1 5 2 0 6 has 2 rows and 3 columns and so it is a 2 3 matrix (ii 1 0 7 1 2 3 3 1 is a
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 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 informationLinear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)
MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of
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 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 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 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 real-valued matrix A is said to be an orthogonal
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 two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation
More informationBrief Introduction to Vectors and Matrices
CHAPTER 1 Brief Introduction to Vectors and Matrices In this chapter, we will discuss some needed concepts found in introductory course in linear algebra. We will introduce matrix, vector, vector-valued
More informationMATH 551 - APPLIED MATRIX THEORY
MATH 55 - APPLIED MATRIX THEORY FINAL TEST: SAMPLE with SOLUTIONS (25 points NAME: PROBLEM (3 points A web of 5 pages is described by a directed graph whose matrix is given by A Do the following ( points
More informationTHE DIMENSION OF A VECTOR SPACE
THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field
More informationMath 312 Homework 1 Solutions
Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please
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 informationv w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed 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 informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
More informationOrthogonal Diagonalization of Symmetric Matrices
MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding
More information6. LECTURE 6. Objectives
6. LECTURE 6 Objectives I understand how to use vectors to understand displacement. I can find the magnitude of a vector. I can sketch a vector. I can add and subtract vector. I can multiply a vector by
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationAlgebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.
More informationReview of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools
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 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 n-dimensional column
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 informationName: Section Registered In:
Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are
More informationMath Common Core Sampler Test
High School Algebra Core Curriculum Math Test Math Common Core Sampler Test Our High School Algebra sampler covers the twenty most common questions that we see targeted for this level. For complete tests
More informationSolving Systems of Linear Equations
LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how
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 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 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 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 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 information5 Homogeneous systems
5 Homogeneous systems Definition: A homogeneous (ho-mo-jeen -i-us) system of linear algebraic equations is one in which all the numbers on the right hand side are equal to : a x +... + a n x n =.. a m
More informationMatrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.
Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that
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 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 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 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 informationPrecalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES
Content Expectations for Precalculus Michigan Precalculus 2011 REVERSE CORRELATION CHAPTER/LESSON TITLES Chapter 0 Preparing for Precalculus 0-1 Sets There are no state-mandated Precalculus 0-2 Operations
More informationWe shall turn our attention to solving linear systems of equations. Ax = b
59 Linear Algebra We shall turn our attention to solving linear systems of equations Ax = b where A R m n, x R n, and b R m. We already saw examples of methods that required the solution of a linear system
More information1 2 3 1 1 2 x = + x 2 + x 4 1 0 1
(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which
More informationFactoring Polynomials
Factoring Polynomials Hoste, Miller, Murieka September 12, 2011 1 Factoring In the previous section, we discussed how to determine the product of two or more terms. Consider, for instance, the equations
More information[1] Diagonal factorization
8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:
More informationDRAFT. Further mathematics. GCE AS and A level subject content
Further mathematics GCE AS and A level subject content July 2014 s Introduction Purpose Aims and objectives Subject content Structure Background knowledge Overarching themes Use of technology Detailed
More informationThe Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
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 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 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 x-axis points out
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 informationTable of Contents. Depth of Knowledge p. 12 Introduction: Numbers and Number Systems p. 13
AERO MATHEMATICS CURRICULUM FRAMEWORK HIGH SCHOOL STANDARDS Adopted from the Common Core Standards Table of Contents Introduction p. 4 Depth of Knowledge p. 12 Introduction: Numbers and Number Systems
More informationLinear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
More informationVector Spaces 4.4 Spanning and Independence
Vector Spaces 4.4 and Independence October 18 Goals Discuss two important basic concepts: Define linear combination of vectors. Define Span(S) of a set S of vectors. Define linear Independence of a set
More informationProblem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.
Math 312, Fall 2012 Jerry L. Kazdan Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. In addition to the problems below, you should also know how to solve
More informationCORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
More informationLEARNING OBJECTIVES FOR THIS CHAPTER
CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. Finite-Dimensional
More information1 0 5 3 3 A = 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0
Solutions: Assignment 4.. Find the redundant column vectors of the given matrix A by inspection. Then find a basis of the image of A and a basis of the kernel of A. 5 A The second and third columns are
More informationMATH1231 Algebra, 2015 Chapter 7: Linear maps
MATH1231 Algebra, 2015 Chapter 7: Linear maps A/Prof. Daniel Chan School of Mathematics and Statistics University of New South Wales danielc@unsw.edu.au Daniel Chan (UNSW) MATH1231 Algebra 1 / 43 Chapter
More informationAlgebraic Concepts Algebraic Concepts Writing
Curriculum Guide: Algebra 2/Trig (AR) 2 nd Quarter 8/7/2013 2 nd Quarter, Grade 9-12 GRADE 9-12 Unit of Study: Matrices Resources: Textbook: Algebra 2 (Holt, Rinehart & Winston), Ch. 4 Length of Study:
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 information