LESSON 35: EIGENVALUES AND EIGENVECTORS APRIL 21, (1) We might also write v as v. Both notations refer to a vector.

Size: px
Start display at page:

Download "LESSON 35: EIGENVALUES AND EIGENVECTORS APRIL 21, (1) We might also write v as v. Both notations refer to a vector."

Transcription

1 LESSON 5: EIGENVALUES AND EIGENVECTORS APRIL 2, 27 In this contet, a vector is a column matri E Note 2 v 2, v () We might also write v as v Both notations refer to a vector (2) A vector can be man different things depending on the contet A vector might be a row matri or even a function Here, a column matri will suffice Given a matri A, there alwas eists a number λ and a collection of non-zero vectors v λ such that A v λ λ v λ This means that multipling v λ b A doesn t reall change the essence of v λ Definition We sa λ is an eigenvalue and v λ is an eigenvector associated to λ Note 4 In German, eigen means something like same So this is sort of like saing that this vector isn t essentiall changed b multipling b A Remark 5 The zero vector (ie, the column vectors with all zeros denoted or ) does not count as an eigenvector because A for all matrices A and so it doesn t give ou an information about the matri itself 2 E 6 Let A Then Eigenvalue One Associated Eigenvector λ 2 v λ 2 v 2 But there are man more eigenvectors for each eigenvalue In fact, all eigenvalues associated to λ 2 are of the form t w

2 2 MATH 62 for some t and all eigenvectors associated to λ 2 are of the form w 2 t for some t Eercise Check that and Now, the question is: how do ou find eigenvalues and eigenvectors? a b Let A and let I (I is the 2 2 identit matri) c d Fact 7 If λ is an eigenvalue of A, then λi A is a singular matri Recall that a singular matri is a matri whose determinant is zero So, to find the eigenvalues of A, we need to find the λ such that det(λi A) First, let s figure out what λi A is We write: a b λi A λ c d λ a b λ c d λ a b c λ d Then det(λi A) λ a b c λ d (λ a)(λ d) ( b)( c) (λ a)(λ d) bc This means we need to find the λ such that Eample Let A 2 (λ a)(λ d) bc Find the eigenvalues of A

3 MATH 62 Solution: If λ is an eigenvalue of A, then det(λi A) So, det(λi A) λ 2 λ (λ )(λ) ()( 2) λ 2 λ + 2 Therefore, our eigenvalues are λ, 2 (λ )(λ 2) After we have found an eigenvalue, we want to find the associated eigenvectors Eample 2 Let A be as in the eample above Find an eigenvector for each eigenvalue of A Solution: We know that the eigenvalues of A are λ, 2 We find the associated eigenvectors for λ first and then λ 2 λ : Assume that is an eigenvector associated to λ Then, b the definition of an eigenvector, we must have λ 2 We want to find a relation on and that will make this true So, we multipl the matrices on the left: () + ( )() 2 2() + () 2 B the equation above, we must have 2 This gives use two equations: 2 We see that both of these equations give the same information: 2 So, 2 describes the eigenvectors associated to λ For a specific eigenvector associated to λ, we can take to be an number Sa Then 2 is an eigenvector associated to λ

4 4 MATH 62 Eercise 2 Check that Note 8 If ou had written and then put 2 2, is also true Both 2 and 2 describe the collection of eigenvectors associated to λ λ 2: Let be an eigenvector associated to λ 2 Then the following must be true: 2 2 Using matri multiplication on the left, we get This again gives us two equations: B the second equation, we see that Hence, the eigenvectors associated to λ 2 look like or For a specific eigenvector, we might take Thus is an eigenvector associated to λ 2 Eercise Check that 2 2 Eample Let A Find the eigenvalues of A and for each eigenvalue find an associated eigenvector Solution: We alwas start b finding the eigenvalues of A

5 MATH 62 5 Eigenvalues: Write det(λi A) λ λ λ λ (λ )(λ ) ( )() λ 2 4λ + + λ 2 4λ + 4 (λ 2) 2 λ 2 is a repeated eigenvalue, which is fine This actuall makes our work easier because now we onl need to find the eigenvectors for one eigenvalue Eigenvectors: Let be an eigenvector for λ 2 Then Using matri multiplication on the left, we get So our equations are: From this, we get that Hence, eigenvectors associated to λ 2 are of the form Eample 4 Check if either v 5 or w are eigenvectors of A If so, find the associated eigenvalue Solution: The eas wa to do this is to simpl multipl A b v and w and see what happens

6 6 MATH 62 Let s start with A v We write 2 5 A v 4 2( ) + 5() 4( ) + ( )() ( ) 7() 7 7 v Therefore, v is an eigenvector associated to λ 7 Net, we check w Write A w 4 2(5) + 5(5) 4(5) + ( )() Now, the question is: does there eists a λ such that 5 5 5λ λ λ If there did, we would get 5 5λ λ But the first equation implies λ and the second implies λ So no such λ eists Therefore, w is not an eigenvector of A Eercise 4 Show that λ 2 is the other eigenvalue of A Eample 5 Find the eigenvalues and eigenvectors of 2 5 A 2 Solution: We start with the eigenvalues

7 MATH 62 7 Eigenvalues: Since we write λi A λ 2 5 λ + 2 det(λi A) λ 2 5 λ + 2, (λ 2)(λ + 2) ( )( 5) λ λ 2 9 (λ )(λ + ) Therefore, our eigenvalues are λ, Eigenvectors: Because we have two eigenvalues, we need to break this into cases: λ : Let be an eigenvector associated to λ Then λ Using matri multiplication on the left and scalar multiplication on the right, we get The equations we get from this are: Solving, we find Hence, the eigenvectors associated to λ are of the form A specific eigenvector can be found b taking : λ : Let be an eigenvector associated to λ Then we must have Simplifing on the left and right, this equation becomes

8 8 MATH 62 So we get the equations Hence, 5 So eigenvectors associated to λ are of the form 5 5 A specific eigenvector can be obtained b taking :

Eigenvectors and Eigenvalues 1

Eigenvectors and Eigenvalues 1 Ma 2015 page 1 Eigenvectors and Eigenvalues 1 In this handout, we will eplore eigenvectors and eigenvalues. We will begin with an eploration, then provide some direct eplanation and worked eamples, and

More information

15. Eigenvalues, Eigenvectors

15. Eigenvalues, Eigenvectors 5 Eigenvalues, Eigenvectors Matri of a Linear Transformation Consider a linear ( transformation ) L : a b R 2 R 2 Suppose we know that L and L Then c d because of linearit, we can determine what L does

More information

Computer Graphics: 2D Transformations. Course Website:

Computer Graphics: 2D Transformations. Course Website: Computer Graphics: D Transformations Course Website: http://www.comp.dit.ie/bmacnamee 5 Contents Wh transformations Transformations Translation Scaling Rotation Homogeneous coordinates Matri multiplications

More information

Mathematics 309 Conic sections and their applicationsn. Chapter 2. Quadric figures. ai,j x i x j + b i x i + c =0. 1. Coordinate changes

Mathematics 309 Conic sections and their applicationsn. Chapter 2. Quadric figures. ai,j x i x j + b i x i + c =0. 1. Coordinate changes Mathematics 309 Conic sections and their applicationsn Chapter 2. Quadric figures In this chapter want to outline quickl how to decide what figure associated in 2D and 3D to quadratic equations look like.

More information

Some linear transformations on R 2 Math 130 Linear Algebra D Joyce, Fall 2013

Some linear transformations on R 2 Math 130 Linear Algebra D Joyce, Fall 2013 Some linear transformations on R 2 Math 3 Linear Algebra D Joce, Fall 23 Let s look at some some linear transformations on the plane R 2. We ll look at several kinds of operators on R 2 including reflections,

More information

MATRIX TRANSFORMATIONS

MATRIX TRANSFORMATIONS CHAPTER 5. MATRIX TRANSFORMATIONS INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW MATRIX TRANSFORMATIONS Matri Transformations Definition Let A and B be sets. A function f : A B

More information

MAT1302F Mathematical Methods II Lecture 19

MAT1302F Mathematical Methods II Lecture 19 MAT302F Mathematical Methods II Lecture 9 Aaron Christie 2 April 205 Eigenvectors, Eigenvalues, and Diagonalization Now that the basic theory of eigenvalues and eigenvectors is in place most importantly

More information

LU Factorization. A m x n matrix A admits an LU factorization if it can be written in the form of A = LU

LU Factorization. A m x n matrix A admits an LU factorization if it can be written in the form of A = LU LU Factorization A m n matri A admits an LU factorization if it can be written in the form of Where, A = LU L : is a m m lower triangular matri with s on the diagonal. The matri L is invertible and is

More information

Math-2. Lesson:1-2 Properties of Exponents

Math-2. Lesson:1-2 Properties of Exponents Math- Lesson:- Properties of Eponents Properties of Eponents What is a power? Power: An epression formed b repeated multiplication of the same factor. Coefficient Base Eponent The eponent applies to the

More information

Math Lesson 2-2 Properties of Exponents

Math Lesson 2-2 Properties of Exponents Math-00 Lesson - Properties of Eponents Properties of Eponents What is a power? Power: An epression formed b repeated multiplication of the base. Coefficient Base Eponent The eponent applies to the number

More information

6. Linear transformations. Consider the function. f : R 2 R 2 which sends (x, y) (x, y)

6. Linear transformations. Consider the function. f : R 2 R 2 which sends (x, y) (x, y) Consider the function 6 Linear transformations f : R 2 R 2 which sends (x, ) (, x) This is an example of a linear transformation Before we get into the definition of a linear transformation, let s investigate

More information

Introduction to Vector Spaces Linear Algebra, Spring 2011

Introduction to Vector Spaces Linear Algebra, Spring 2011 Introduction to Vector Spaces Linear Algebra, Spring 2011 You probabl have heard the word vector before, perhaps in the contet of Calculus III or phsics. You probabl think of a vector like this: 5 3 or

More information

Unit 12 Study Notes 1 Systems of Equations

Unit 12 Study Notes 1 Systems of Equations You should learn to: Unit Stud Notes Sstems of Equations. Solve sstems of equations b substitution.. Solve sstems of equations b graphing (calculator). 3. Solve sstems of equations b elimination. 4. Solve

More information

ES.1803 Topic 16 Notes Jeremy Orloff

ES.1803 Topic 16 Notes Jeremy Orloff ES803 Topic 6 Notes Jerem Orloff 6 Eigenalues, diagonalization, decoupling This note coers topics that will take us seeral classes to get through We will look almost eclusiel at 2 2 matrices These hae

More information

Mathematics of Cryptography Part I

Mathematics of Cryptography Part I CHAPTER 2 Mathematics of Crptograph Part I (Solution to Practice Set) Review Questions 1. The set of integers is Z. It contains all integral numbers from negative infinit to positive infinit. The set of

More information

JUST THE MATHS SLIDES NUMBER 9.6. MATRICES 6 (Eigenvalues and eigenvectors) A.J.Hobson

JUST THE MATHS SLIDES NUMBER 9.6. MATRICES 6 (Eigenvalues and eigenvectors) A.J.Hobson JUST THE MATHS SLIDES NUMBER 96 MATRICES 6 (Eigenvalues and eigenvectors) by AJHobson 96 The statement of the problem 962 The solution of the problem UNIT 96 - MATRICES 6 EIGENVALUES AND EIGENVECTORS 96

More information

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors 5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION DETERMINANATS nn Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling),

More information

Exact Equations. M(x,y) + N(x,y) y = 0, M(x,y) dx + N(x,y) dy = 0. M(x,y) + N(x,y) y = 0

Exact Equations. M(x,y) + N(x,y) y = 0, M(x,y) dx + N(x,y) dy = 0. M(x,y) + N(x,y) y = 0 Eact Equations An eact equation is a first order differential equation that can be written in the form M(, + N(,, provided that there eists a function ψ(, such that = M (, and N(, = Note : Often the equation

More information

Homework Notes Week 6

Homework Notes Week 6 Homework Notes Week 6 Math 24 Spring 24 34#4b The sstem + 2 3 3 + 4 = 2 + 2 + 3 4 = 2 + 2 3 = is consistent To see this we put the matri 3 2 A b = 2 into reduced row echelon form Adding times the first

More information

Matrices. VCE Maths Methods - Unit 2 - Matrices

Matrices. VCE Maths Methods - Unit 2 - Matrices Matrices Introduction to matrices Addition & subtraction Scalar multiplication Matri multiplication The unit matri Matri division - the inverse matri Using matrices - simultaneous equations Matri transformations

More information

12.1 Systems of Linear equations: Substitution and Elimination

12.1 Systems of Linear equations: Substitution and Elimination . Sstems of Linear equations: Substitution and Elimination Sstems of two linear equations in two variables A sstem of equations is a collection of two or more equations. A solution of a sstem in two variables

More information

Matrices. VCE Maths Methods - Unit 2 - Matrices

Matrices. VCE Maths Methods - Unit 2 - Matrices Matrices Introduction to matrices Addition subtraction Scalar multiplication Matri multiplication The unit matri Matri division - the inverse matri Using matrices - simultaneous equations Matri transformations

More information

Eigenvalue and Eigenvector Homework

Eigenvalue and Eigenvector Homework Eigenvalue and Eigenvector Homework Olena Bormashenko November 4, 2 For each of the matrices A below, do the following:. Find the characteristic polynomial of A, and use it to find all the eigenvalues

More information

Affine transformations

Affine transformations Reading Optional reading: Affine transformations Brian Curless CSE 557 Autumn 207 Angel and Shreiner: 3., 3.7-3. Marschner and Shirle: 2.3, 2.4.-2.4.4, 6..-6..4, 6.2., 6.3 Further reading: Angel, the rest

More information

1 GSW Gaussian Elimination

1 GSW Gaussian Elimination Gaussian elimination is probabl the simplest technique for solving a set of simultaneous linear equations, such as: = A x + A x + A x +... + A x,,,, n n = A x + A x + A x +... + A x,,,, n n... m = Am,x

More information

Identifying second degree equations

Identifying second degree equations Chapter 7 Identifing second degree equations 71 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +

More information

ES.182A Topic 41 Notes Jeremy Orloff. 41 Extensions and applications of Green s theorem

ES.182A Topic 41 Notes Jeremy Orloff. 41 Extensions and applications of Green s theorem ES.182A Topic 41 Notes Jerem Orloff 41 Etensions and applications of Green s theorem 41.1 eview of Green s theorem: Tangential (work) form: F T ds = curlf d d M d + N d = N M d d. Normal (flu) form: F

More information

Ch 3 Alg 2 Note Sheet.doc 3.1 Graphing Systems of Equations

Ch 3 Alg 2 Note Sheet.doc 3.1 Graphing Systems of Equations Ch 3 Alg Note Sheet.doc 3.1 Graphing Sstems of Equations Sstems of Linear Equations A sstem of equations is a set of two or more equations that use the same variables. If the graph of each equation =.4

More information

Eigenvalues and Eigenvectors 7.1 Eigenvalues and Eigenvecto

Eigenvalues and Eigenvectors 7.1 Eigenvalues and Eigenvecto 7.1 November 6 7.1 Eigenvalues and Eigenvecto Goals Suppose A is square matrix of order n. Eigenvalues of A will be defined. Eigenvectors of A, corresponding to each eigenvalue, will be defined. Eigenspaces

More information

Dimension. Eigenvalue and eigenvector

Dimension. Eigenvalue and eigenvector Dimension. Eigenvalue and eigenvector Math 112, week 9 Goals: Bases, dimension, rank-nullity theorem. Eigenvalue and eigenvector. Suggested Textbook Readings: Sections 4.5, 4.6, 5.1, 5.2 Week 9: Dimension,

More information

UNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives

UNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives Chapter 3 3Quadratics Objectives To recognise and sketch the graphs of quadratic polnomials. To find the ke features of the graph of a quadratic polnomial: ais intercepts, turning point and ais of smmetr.

More information

1.7 Inverse Functions

1.7 Inverse Functions 71_0107.qd 1/7/0 10: AM Page 17 Section 1.7 Inverse Functions 17 1.7 Inverse Functions Inverse Functions Recall from Section 1. that a function can be represented b a set of ordered pairs. For instance,

More information

Chapter Adequacy of Solutions

Chapter Adequacy of Solutions Chapter 04.09 dequac of Solutions fter reading this chapter, ou should be able to: 1. know the difference between ill-conditioned and well-conditioned sstems of equations,. define the norm of a matri,

More information

Affine transformations

Affine transformations Reading Required: Affine transformations Brian Curless CSE 557 Fall 2009 Shirle, Sec. 2.4, 2.7 Shirle, Ch. 5.-5.3 Shirle, Ch. 6 Further reading: Fole, et al, Chapter 5.-5.5. David F. Rogers and J. Alan

More information

Unit 3 NOTES Honors Common Core Math 2 1. Day 1: Properties of Exponents

Unit 3 NOTES Honors Common Core Math 2 1. Day 1: Properties of Exponents Unit NOTES Honors Common Core Math Da : Properties of Eponents Warm-Up: Before we begin toda s lesson, how much do ou remember about eponents? Use epanded form to write the rules for the eponents. OBJECTIVE

More information

Affine transformations. Brian Curless CSE 557 Fall 2014

Affine transformations. Brian Curless CSE 557 Fall 2014 Affine transformations Brian Curless CSE 557 Fall 2014 1 Reading Required: Shirle, Sec. 2.4, 2.7 Shirle, Ch. 5.1-5.3 Shirle, Ch. 6 Further reading: Fole, et al, Chapter 5.1-5.5. David F. Rogers and J.

More information

Chapter 5: Systems of Equations

Chapter 5: Systems of Equations Chapter : Sstems of Equations Section.: Sstems in Two Variables... 0 Section. Eercises... 9 Section.: Sstems in Three Variables... Section. Eercises... Section.: Linear Inequalities... Section.: Eercises.

More information

Chapter 18 Quadratic Function 2

Chapter 18 Quadratic Function 2 Chapter 18 Quadratic Function Completed Square Form 1 Consider this special set of numbers - the square numbers or the set of perfect squares. 4 = = 9 = 3 = 16 = 4 = 5 = 5 = Numbers like 5, 11, 15 are

More information

Linear Equations in Linear Algebra

Linear Equations in Linear Algebra 1 Linear Equations in Linear Algebra 1.1 SYSTEMS OF LINEAR EQUATIONS LINEAR EQUATION,, 1 n A linear equation in the variables equation that can be written in the form a a a b 1 1 2 2 n n a a is an where

More information

Demonstrate solution methods for systems of linear equations. Show that a system of equations can be represented in matrix-vector form.

Demonstrate solution methods for systems of linear equations. Show that a system of equations can be represented in matrix-vector form. Chapter Linear lgebra Objective Demonstrate solution methods for sstems of linear equations. Show that a sstem of equations can be represented in matri-vector form. 4 Flowrates in kmol/hr Figure.: Two

More information

Intro to Nonlinear Optimization

Intro to Nonlinear Optimization Intro to Nonlinear Optimization We now rela the proportionality and additivity assumptions of LP What are the challenges of nonlinear programs NLP s? Objectives and constraints can use any function: ma

More information

Math 214 Spring problem set (a) Consider these two first order equations. (I) dy dx = x + 1 dy

Math 214 Spring problem set (a) Consider these two first order equations. (I) dy dx = x + 1 dy Math 4 Spring 08 problem set. (a) Consider these two first order equations. (I) d d = + d (II) d = Below are four direction fields. Match the differential equations above to their direction fields. Provide

More information

CS 378: Computer Game Technology

CS 378: Computer Game Technology CS 378: Computer Game Technolog 3D Engines and Scene Graphs Spring 202 Universit of Teas at Austin CS 378 Game Technolog Don Fussell Representation! We can represent a point, p =,), in the plane! as a

More information

Section 3.1. ; X = (0, 1]. (i) f : R R R, f (x, y) = x y

Section 3.1. ; X = (0, 1]. (i) f : R R R, f (x, y) = x y Paul J. Bruillard MATH 0.970 Problem Set 6 An Introduction to Abstract Mathematics R. Bond and W. Keane Section 3.1: 3b,c,e,i, 4bd, 6, 9, 15, 16, 18c,e, 19a, 0, 1b Section 3.: 1f,i, e, 6, 1e,f,h, 13e,

More information

RELATIONS AND FUNCTIONS through

RELATIONS AND FUNCTIONS through RELATIONS AND FUNCTIONS 11.1.2 through 11.1. Relations and Functions establish a correspondence between the input values (usuall ) and the output values (usuall ) according to the particular relation or

More information

and let s calculate the image of some vectors under the transformation T.

and let s calculate the image of some vectors under the transformation T. Chapter 5 Eigenvalues and Eigenvectors 5. Eigenvalues and Eigenvectors Let T : R n R n be a linear transformation. Then T can be represented by a matrix (the standard matrix), and we can write T ( v) =

More information

Properties of Linear Transformations from R n to R m

Properties of Linear Transformations from R n to R m Properties of Linear Transformations from R n to R m MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Topic Overview Relationship between the properties of a matrix transformation

More information

Systems of Linear Equations: Solving by Graphing

Systems of Linear Equations: Solving by Graphing 8.1 Sstems of Linear Equations: Solving b Graphing 8.1 OBJECTIVE 1. Find the solution(s) for a set of linear equations b graphing NOTE There is no other ordered pair that satisfies both equations. From

More information

Definition: div Let n, d 0. We define ndiv d as the least integer quotient obtained when n is divided by d. That is if

Definition: div Let n, d 0. We define ndiv d as the least integer quotient obtained when n is divided by d. That is if Section 5. Congruence Arithmetic A number of computer languages have built-in functions that compute the quotient and remainder of division. Definition: div Let n, d 0. We define ndiv d as the least integer

More information

Chapter 12: Iterative Methods

Chapter 12: Iterative Methods ES 40: Scientific and Engineering Computation. Uchechukwu Ofoegbu Temple University Chapter : Iterative Methods ES 40: Scientific and Engineering Computation. Gauss-Seidel Method The Gauss-Seidel method

More information

Section 1.2: A Catalog of Functions

Section 1.2: A Catalog of Functions Section 1.: A Catalog of Functions As we discussed in the last section, in the sciences, we often tr to find an equation which models some given phenomenon in the real world - for eample, temperature as

More information

State space transformations

State space transformations Capitolo 0. INTRODUCTION. State space transformations Let us consider the following linear time-invariant system: { ẋ(t) = A(t)+Bu(t) y(t) = C(t)+Du(t) () A state space transformation can be obtained using

More information

Are You Ready? Find Area in the Coordinate Plane

Are You Ready? Find Area in the Coordinate Plane SKILL 38 Are You Read? Find Area in the Coordinate Plane Teaching Skill 38 Objective Find the areas of figures in the coordinate plane. Review with students the definition of area. Ask: Is the definition

More information

Intermediate Algebra. Gregg Waterman Oregon Institute of Technology

Intermediate Algebra. Gregg Waterman Oregon Institute of Technology Intermediate Algebra Gregg Waterman Oregon Institute of Technolog c 2017 Gregg Waterman This work is licensed under the Creative Commons Attribution 4.0 International license. The essence of the license

More information

A Note on the Eigenvalues and Eigenvectors of Leslie matrices. Ralph Howard Department of Mathematics University of South Carolina

A Note on the Eigenvalues and Eigenvectors of Leslie matrices. Ralph Howard Department of Mathematics University of South Carolina A Note on the Eigenvalues and Eigenvectors of Leslie matrices Ralph Howard Department of Mathematics University of South Carolina Vectors and Matrices A size n vector, v, is a list of n numbers put in

More information

Handout for Adequacy of Solutions Chapter SET ONE The solution to Make a small change in the right hand side vector of the equations

Handout for Adequacy of Solutions Chapter SET ONE The solution to Make a small change in the right hand side vector of the equations Handout for dequac of Solutions Chapter 04.07 SET ONE The solution to 7.999 4 3.999 Make a small change in the right hand side vector of the equations 7.998 4.00 3.999 4.000 3.999 Make a small change in

More information

11.4 Polar Coordinates

11.4 Polar Coordinates 11. Polar Coordinates 917 11. Polar Coordinates In Section 1.1, we introduced the Cartesian coordinates of a point in the plane as a means of assigning ordered pairs of numbers to points in the plane.

More information

14.1 Systems of Linear Equations in Two Variables

14.1 Systems of Linear Equations in Two Variables 86 Chapter 1 Sstems of Equations and Matrices 1.1 Sstems of Linear Equations in Two Variables Use the method of substitution to solve sstems of equations in two variables. Use the method of elimination

More information

REVISION SHEET FP2 (MEI) CALCULUS. x x 0.5. x x 1.5. π π. Standard Calculus of Inverse Trig and Hyperbolic Trig Functions = + = + arcsin x = +

REVISION SHEET FP2 (MEI) CALCULUS. x x 0.5. x x 1.5. π π. Standard Calculus of Inverse Trig and Hyperbolic Trig Functions = + = + arcsin x = + the Further Mathematics network www.fmnetwork.org.uk V 07 REVISION SHEET FP (MEI) CALCULUS The main ideas are: Calculus using inverse trig functions & hperbolic trig functions and their inverses. Maclaurin

More information

Recall : Eigenvalues and Eigenvectors

Recall : Eigenvalues and Eigenvectors Recall : Eigenvalues and Eigenvectors Let A be an n n matrix. If a nonzero vector x in R n satisfies Ax λx for a scalar λ, then : The scalar λ is called an eigenvalue of A. The vector x is called an eigenvector

More information

Math 4242 Fall 2016 (Darij Grinberg): homework set 6 due: Mon, 21 Nov 2016 Let me first recall a definition.

Math 4242 Fall 2016 (Darij Grinberg): homework set 6 due: Mon, 21 Nov 2016 Let me first recall a definition. Math 4242 Fall 206 homework page Math 4242 Fall 206 Darij Grinberg: homework set 6 due: Mon, 2 Nov 206 Let me first recall a definition. Definition 0.. Let V and W be two vector spaces. Let v = v, v 2,...,

More information

Math 123 Summary of Important Algebra & Trigonometry Concepts Chapter 1 & Appendix D, Stewart, Calculus Early Transcendentals

Math 123 Summary of Important Algebra & Trigonometry Concepts Chapter 1 & Appendix D, Stewart, Calculus Early Transcendentals Math Summar of Important Algebra & Trigonometr Concepts Chapter & Appendi D, Stewart, Calculus Earl Transcendentals Function a rule that assigns to each element in a set D eactl one element, called f (

More information

Two conventions for coordinate systems. Left-Hand vs Right-Hand. x z. Which is which?

Two conventions for coordinate systems. Left-Hand vs Right-Hand. x z. Which is which? walters@buffalo.edu CSE 480/580 Lecture 2 Slide 3-D Transformations 3-D space Two conventions for coordinate sstems Left-Hand vs Right-Hand (Thumb is the ais, inde is the ais) Which is which? Most graphics

More information

2009 Math Olympics Level I

2009 Math Olympics Level I Saginaw Valle State Universit 009 Math Olmpics Level I. A man and his wife take a trip that usuall takes three hours if the drive at an average speed of 60 mi/h. After an hour and a half of driving at

More information

MATH Line integrals III Fall The fundamental theorem of line integrals. In general C

MATH Line integrals III Fall The fundamental theorem of line integrals. In general C MATH 255 Line integrals III Fall 216 In general 1. The fundamental theorem of line integrals v T ds depends on the curve between the starting point and the ending point. onsider two was to get from (1,

More information

Math 369 Exam #1 Practice Problems

Math 369 Exam #1 Practice Problems Math 69 Exam # Practice Problems Find the set of solutions of the following sstem of linear equations Show enough work to make our steps clear x + + z + 4w x 4z 6w x + 5 + 7z + w Answer: We solve b forming

More information

Matrices and Determinants

Matrices and Determinants Math Assignment Eperts is a leading provider of online Math help. Our eperts have prepared sample assignments to demonstrate the quality of solution we provide. If you are looking for mathematics help

More information

NAME MATH 304 Examination 2 Page 1

NAME MATH 304 Examination 2 Page 1 NAME MATH 4 Examination 2 Page. [8 points (a) Find the following determinant. However, use only properties of determinants, without calculating directly (that is without expanding along a column or row

More information

Vector and Affine Math

Vector and Affine Math Vector and Affine Math Computer Science Department The Universit of Teas at Austin Vectors A vector is a direction and a magnitude Does NOT include a point of reference Usuall thought of as an arrow in

More information

Green s Theorem Jeremy Orloff

Green s Theorem Jeremy Orloff Green s Theorem Jerem Orloff Line integrals and Green s theorem. Vector Fields Vector notation. In 8.4 we will mostl use the notation (v) = (a, b) for vectors. The other common notation (v) = ai + bj runs

More information

3.7 InveRSe FUnCTIOnS

3.7 InveRSe FUnCTIOnS CHAPTER functions learning ObjeCTIveS In this section, ou will: Verif inverse functions. Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.

More information

Review Topics for MATH 1400 Elements of Calculus Table of Contents

Review Topics for MATH 1400 Elements of Calculus Table of Contents Math 1400 - Mano Table of Contents - Review - page 1 of 2 Review Topics for MATH 1400 Elements of Calculus Table of Contents MATH 1400 Elements of Calculus is one of the Marquette Core Courses for Mathematical

More information

Worksheet #1. A little review.

Worksheet #1. A little review. Worksheet #1. A little review. I. Set up BUT DO NOT EVALUATE definite integrals for each of the following. 1. The area between the curves = 1 and = 3. Solution. The first thing we should ask ourselves

More information

Linear Algebra Practice Problems

Linear Algebra Practice Problems Math 7, Professor Ramras Linear Algebra Practice Problems () Consider the following system of linear equations in the variables x, y, and z, in which the constants a and b are real numbers. x y + z = a

More information

Eigenvalues & Eigenvectors

Eigenvalues & Eigenvectors Eigenvalues & Eigenvectors Page 1 Eigenvalues are a very important concept in linear algebra, and one that comes up in other mathematics courses as well. The word eigen is German for inherent or characteristic,

More information

8. BOOLEAN ALGEBRAS x x

8. BOOLEAN ALGEBRAS x x 8. BOOLEAN ALGEBRAS 8.1. Definition of a Boolean Algebra There are man sstems of interest to computing scientists that have a common underling structure. It makes sense to describe such a mathematical

More information

Can a system of linear equations have no solution? Can a system of linear equations have many solutions?

Can a system of linear equations have no solution? Can a system of linear equations have many solutions? 5. Solving Special Sstems of Linear Equations Can a sstem of linear equations have no solution? Can a sstem of linear equations have man solutions? ACTIVITY: Writing a Sstem of Linear Equations Work with

More information

Now, if you see that x and y are present in both equations, you may write:

Now, if you see that x and y are present in both equations, you may write: Matrices: Suppose you have two simultaneous equations: y y 3 () Now, if you see that and y are present in both equations, you may write: y 3 () You should be able to see where the numbers have come from.

More information

Reading. Affine transformations. Vector representation. Geometric transformations. x y z. x y. Required: Angel 4.1, Further reading:

Reading. Affine transformations. Vector representation. Geometric transformations. x y z. x y. Required: Angel 4.1, Further reading: Reading Required: Angel 4.1, 4.6-4.10 Further reading: Affine transformations Angel, the rest of Chapter 4 Fole, et al, Chapter 5.1-5.5. David F. Rogers and J. Alan Adams, Mathematical Elements for Computer

More information

MATH 1324 (Finite Mathematics or Business Math I) Lecture Notes Author / Copyright: Kevin Pinegar

MATH 1324 (Finite Mathematics or Business Math I) Lecture Notes Author / Copyright: Kevin Pinegar MTH Finite Mathematics or usiness Math Lecture Notes uthor / opright: Kevin Pinegar MTH Module Notes: SYSTEMS OF EQUTONS & MTES. MT NVESES & POPETES OF MTES Definition: We cannot discuss the inverse of

More information

31. TRANSFORMING TOOL #2 (the Multiplication Property of Equality)

31. TRANSFORMING TOOL #2 (the Multiplication Property of Equality) 3 TRANSFORMING TOOL # (the Multiplication Property of Equality) a second transforming tool THEOREM Multiplication Property of Equality In the previous section, we learned that adding/subtracting the same

More information

5. Zeros. We deduce that the graph crosses the x-axis at the points x = 0, 1, 2 and 4, and nowhere else. And that s exactly what we see in the graph.

5. Zeros. We deduce that the graph crosses the x-axis at the points x = 0, 1, 2 and 4, and nowhere else. And that s exactly what we see in the graph. . Zeros Eample 1. At the right we have drawn the graph of the polnomial = ( 1) ( 2) ( 4). Argue that the form of the algebraic formula allows ou to see right awa where the graph is above the -ais, where

More information

CIVL Statics. Moment of Inertia - Composite Area. A math professor in an unheated room is cold and calculating. Radius of Gyration

CIVL Statics. Moment of Inertia - Composite Area. A math professor in an unheated room is cold and calculating. Radius of Gyration CVL 131 - Statics Moment of nertia Composite Areas A math professor in an unheated room is cold and calculating. Radius of Gration This actuall sounds like some sort of rule for separation on a dance floor.

More information

Get Solution of These Packages & Learn by Video Tutorials on Matrices

Get Solution of These Packages & Learn by Video Tutorials on  Matrices FEE Download Stud Package from website: wwwtekoclassescom & wwwmathsbsuhagcom Get Solution of These Packages & Learn b Video Tutorials on wwwmathsbsuhagcom Matrices An rectangular arrangement of numbers

More information

. This is the Basic Chain Rule. x dt y dt z dt Chain Rule in this context.

. This is the Basic Chain Rule. x dt y dt z dt Chain Rule in this context. Math 18.0A Gradients, Chain Rule, Implicit Dierentiation, igher Order Derivatives These notes ocus on our things: (a) the application o gradients to ind normal vectors to curves suraces; (b) the generaliation

More information

+ = + + = x = + = + = 36x

+ = + + = x = + = + = 36x Ch 5 Alg L Homework Worksheets Computation Worksheet #1: You should be able to do these without a calculator! A) Addition (Subtraction = add the opposite of) B) Multiplication (Division = multipl b the

More information

Math 308 Practice Final Exam Page and vector y =

Math 308 Practice Final Exam Page and vector y = Math 308 Practice Final Exam Page Problem : Solving a linear equation 2 0 2 5 Given matrix A = 3 7 0 0 and vector y = 8. 4 0 0 9 (a) Solve Ax = y (if the equation is consistent) and write the general solution

More information

x y plane is the plane in which the stresses act, yy xy xy Figure 3.5.1: non-zero stress components acting in the x y plane

x y plane is the plane in which the stresses act, yy xy xy Figure 3.5.1: non-zero stress components acting in the x y plane 3.5 Plane Stress This section is concerned with a special two-dimensional state of stress called plane stress. It is important for two reasons: () it arises in real components (particularl in thin components

More information

VECTORS IN THREE DIMENSIONS

VECTORS IN THREE DIMENSIONS 1 CHAPTER 2. BASIC TRIGONOMETRY 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW VECTORS IN THREE DIMENSIONS 1 Vectors in Two Dimensions A vector is an object which has magnitude

More information

CS-184: Computer Graphics. Today

CS-184: Computer Graphics. Today CS-184: Computer Graphics Lecture #3: 2D Transformations Prof. James O Brien Universit of California, Berkele V2006-S-03-1.0 Toda 2D Transformations Primitive Operations Scale, Rotate, Shear, Flip, Translate

More information

Essential Question How can you solve a nonlinear system of equations?

Essential Question How can you solve a nonlinear system of equations? .5 Solving Nonlinear Sstems Essential Question Essential Question How can ou solve a nonlinear sstem of equations? Solving Nonlinear Sstems of Equations Work with a partner. Match each sstem with its graph.

More information

Relations. Functions. Bijection and counting.

Relations. Functions. Bijection and counting. Relations.. and counting. s Given two sets A = {,, } B = {,,, 4} Their A B = {(, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, 4), (, 4), (, 4)} Question: What is the cartesian product of? ( is

More information

Section 8.5 Parametric Equations

Section 8.5 Parametric Equations 504 Chapter 8 Section 8.5 Parametric Equations Man shapes, even ones as simple as circles, cannot be represented as an equation where is a function of. Consider, for eample, the path a moon follows as

More information

INTRODUCTION TO DIOPHANTINE EQUATIONS

INTRODUCTION TO DIOPHANTINE EQUATIONS INTRODUCTION TO DIOPHANTINE EQUATIONS In the earl 20th centur, Thue made an important breakthrough in the stud of diophantine equations. His proof is one of the first eamples of the polnomial method. His

More information

Spring 2019 Exam 2 3/27/19 Time Limit: / Problem Points Score. Total: 280

Spring 2019 Exam 2 3/27/19 Time Limit: / Problem Points Score. Total: 280 Math 307 Spring 2019 Exam 2 3/27/19 Time Limit: / Name (Print): Problem Points Score 1 15 2 20 3 35 4 30 5 10 6 20 7 20 8 20 9 20 10 20 11 10 12 10 13 10 14 10 15 10 16 10 17 10 Total: 280 Math 307 Exam

More information

Polynomial approximation and Splines

Polynomial approximation and Splines Polnomial approimation and Splines 1. Weierstrass approimation theorem The basic question we ll look at toda is how to approimate a complicated function f() with a simpler function P () f() P () for eample,

More information

1 HOMOGENEOUS TRANSFORMATIONS

1 HOMOGENEOUS TRANSFORMATIONS HOMOGENEOUS TRANSFORMATIONS Purpose: The purpose of this chapter is to introduce ou to the Homogeneous Transformation. This simple 4 4 transformation is used in the geometr engines of CAD sstems and in

More information

9. TRANSFORMING TOOL #2 (the Multiplication Property of Equality)

9. TRANSFORMING TOOL #2 (the Multiplication Property of Equality) 9 TRANSFORMING TOOL # (the Multiplication Property of Equality) a second transforming tool THEOREM Multiplication Property of Equality In the previous section, we learned that adding/subtracting the same

More information

Problems for M 10/26:

Problems for M 10/26: Math, Lesieutre Problem set # November 4, 25 Problems for M /26: 5 Is λ 2 an eigenvalue of 2? 8 Why or why not? 2 A 2I The determinant is, which means that A 2I has 6 a nullspace, and so there is an eigenvector

More information

Math 110 Linear Algebra Midterm 2 Review October 28, 2017

Math 110 Linear Algebra Midterm 2 Review October 28, 2017 Math 11 Linear Algebra Midterm Review October 8, 17 Material Material covered on the midterm includes: All lectures from Thursday, Sept. 1st to Tuesday, Oct. 4th Homeworks 9 to 17 Quizzes 5 to 9 Sections

More information