Eigenvectors and Eigenvalues 1
|
|
- Griselda Beasley
- 5 years ago
- Views:
Transcription
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 conclude with a problem set. As alwas, tr to think geometricall and algebraicall. Enjo 1. Consider the transformation matri A = a. Find the image of the points (1, 2) and (3, 2). b. You should have found that ' 1 2 '. ' = 6 ' and 8 suppose we consider the vectors u = 1, 2 and v = 3, 2. One of these vectors is stretched b the matri A. Which one? B how much? ' 3 2 ' = 6 '. Now 4 c. You should have found that the vector v was stretched b a factor of 2. We can write this algebraicall as Av = 2v. The graphs below illustrate how this is different from what happens to the vector u. (1, 2) (6, 8) (6, 8) (3, 2) (6, 4) (1, 2) (6, 4) (3, 2) Now consider the vector w = 1,1. Is this vector stretched b the matri A? B how much? Eplain. Include a picture as part of our eplanation. 2. Let s continue eploring the matri and vectors from the last problem: A = v = 3, 2, and w = 1,1. We have observed that Av = 2v and Aw = 3w. Hence, the image of v is a scalar multiple of v and the image of w is a scalar multiple of w. We sa that v and w are eigenvectors for the matri A with corresponding eigenvalues of 2 and 3. a. Is 9, 6 an eigenvector for A? Is 12, 9 an eigenvector for A? What about 2, 2 or 5, 5? Eplain. ', 1 Adapted from Introduction to Linear Algebra (4 th Edition) b Gilbert Strang.
2 Ma 2015 page 2 3. Consider the matri A =. Our goal in this problem is to prepare for the algebra to help us solve for the eigenvectors and eigenvalues for A. a. Suppose is an eigenvector for A with an eigenvalue of λ. Eplain wh = λ. b. Eplain each of the following algebraic steps: = λ = λ = λ 0 0 λ λ 0 0 λ = λ 2 λ ' ' ' = 0 0 ' c. What is the determinant of the matri 1 λ 2 λ '? How can ou tell? d. The matri 1 λ 2 λ ' maps to 0 0, so its determinant is 0. Use this observation to find λ. Hint: Set up and solve a quadratic equation. e. You should have found that λ 1 = 5 and λ 2 = 3. Eplain wh = 5. f. Find an eigenvector for λ 1 = 5. Hint: Find one possible pair of and that solves the equation = 5. g. Use the same technique to find an eigenvector for λ 2 = 3.
3 Ma 2015 page 3 Eplanation of Eigenvectors and Eigenvalues Now that we understand the basics of dimension, we can look at cool applications of this concept. Take for eample the matri, What is What do ou notice about the solution? Well, we can see that the product algebra, this tpe of vector is special, because it satisfies this equation:. In linear This equation states that there eists some vector, when multiplied b a matri A, results in some scalar multiple of the original vector. The smbol (lambda) represent some real constant, which can even take on a value of zero. We interpret this multiplication as saing after appling a transformation A to the vector, the resulting vector is still in the direction of. We need to find a wa to find these special vectors, called eigenvectors, which depend on the composition of matri A. We then need to figure out how these eigenvectors relate to the lambda value, known as the eigenvalue, for each equation. Even though the eigenvalues depend on the eigenvectors, it is much easier to solve for the eigenvalues first using our knowledge of determinants. Calculating Eigenvalues To calculate an eigenvalue, we need to think about what the solutions of look like. We start out b rewriting this problem as. What does this mean? We want the matri to map points besides the origin to the origin. This means is a tpe of projection, and the determinant of projection matrices is alwas 0. Not onl does a projection matri have a 0 determinant, it also is singular, meaning there are some linearl dependent rows and columns. To calculate the eigenvalues we want to solve the equation matri, this means:. For a 2-b-2 Therefore, to find the eigenvalues, we need to solve this quadratic equation:. Eample: Find the eigenvalues of the matri transformation matri for reflecting over the - ais:. First, we need to find the matri. In this case, we find. We want the determinant of the matri to equal zero, because we want a singular projection matri.
4 Ma 2015 page 4 The solutions to this equation are. Therefore, the eigenvalues of this matri are, meaning there are vectors that satisf the equation. Calculating Eigenvectors Once we have our eigenvalues, we can solve for the vectors that complete the equation. Since we alread have the eigenvalues, let s go back to the equation that helped us solve for the eigenvalues,. What do ou notice about the vector? Since the multiplication maps all points to the origin, is in the null-space of Therefore, to solve for, we must find the null-space of Eample: Find the eigenvectors of. Since we alread have the eigenvalues of When, we need to find the respective null-spaces of, a matri whose null-space satisfies the following transformation equations: From these equations, we realize must be 0, and can take on an value. Therefore, the eigenvector of A that corresponds to the eigenvalue 1 is, where. Even though there are infinitel man vectors that would satisf the equation, we sa that there is one unique eigenvector corresponding to the eigenvalue (all point in the same direction). Solving for the eigenvector when the equations a matri whose null-space satisfies The eigenvector of A that corresponds to the eigenvalue -1 is, where. Summar: The matri has 2 eigenvalues and 2 distinct eigenvectors:
5 Ma 2015 page 5 Problem Set 1. In our eample from this handout, we saw that, meaning for matri there was an eigenvector corresponding to the eigenvalue 2. Suppose a. What is b. Is still an eigenvector of A? If so, what is its associated eigenvalue? c. Would an multiple of still be an eigenvector of A? Eplain. a. Find the eigenvalues of A. b. Find the eigenvectors that correspond to the eigenvalues of A. c. Find unit vectors in the direction of the eigenvectors of A. (Normall, mathematicians describe eigenvectors as unit vectors so that it is easier to distinguish unique eigenvectors). d. What is the sum of the eigenvalues? How does that compare with the sum of the diagonal entries of this matri ( )? e. What is the determinant of A? How does this value compare to the product of the eigenvalues? Let s continue with the A from the previous problem. a. How do the eigenvalues and eigenvectors of compare to those of b. How do the eigenvalues and eigenvectors of compare to those of c. How do the eigenvalues and eigenvectors of compare to those of 4. Suppose we have a matri a. What is the maimum number of possible eigenvalues? Eplain b. Optional challenge: Assuming this matri is not a scalar multiple of the identit matri, what is the maimum number of distinct eigenvectors for this matri? c. In what situations would we onl have one eigenvalue? d. Optional challenge: In what situations would we onl have one eigenvector? e. Prove that the sum of the diagonal entries of this matri, known as the trace of the matri, is equal to the sum of the eigenvalues (this fact is true for all square matrices). f. Prove that the determinant of this matri is equal to the product of the eigenvalues (this is true for all matrices).
6 Ma 2015 page 6 5. a. What are the eigenvalues of this matri? b. You should have noticed the eigenvalues of this triangular matri are the elements on the diagonal. Eplain wh this result makes sense. c. What are the eigenvectors of this matri? 6. How do manipulate the equation to prove the results we have found in question 2? (hint: with a scalar k multiplied b two matrices A and B, kab=(ka)b=a(kb) ) a. is an eigenvalue of, as we discovered in 2a. b. is an eigenvalue of, as we discovered in 2b. c. is an eigenvalue of, as we discovered in 2c. 7. Suppose a. Solve b the quadratic formula to find b. What transformation does the matri Q perform? c. Can Q ever have real eigenvalues? Eplain when this occurs. d. Find the eigenvectors of Q when it has real eigenvalues. e. Optional Challenge: Find the eigenvectors of Q when the eigenvalues are comple. 8. The eigenvalues of are equal to the eigenvalues of. a. Eplain this fact using the ideas regarding the relationship between trace/determinant and eigenvalues. b. Show b eample that the eigenvectors of and are not the same. 9. (same matri from 1), a. Do the eigenvalues of A+B equal the eigenvalues of A plus the eigenvalues of B? b. Do the eigenvalues of AB equal the product of the eigenvalues of A and B?
LESSON 35: EIGENVALUES AND EIGENVECTORS APRIL 21, (1) We might also write v as v. Both notations refer to a vector.
LESSON 5: EIGENVALUES AND EIGENVECTORS APRIL 2, 27 In this contet, a vector is a column matri E Note 2 v 2, v 4 5 6 () We might also write v as v Both notations refer to a vector (2) A vector can be man
More information3.7 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 information15. 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 informationFunctions. Introduction
Functions,00 P,000 00 0 70 7 80 8 0 000 00 00 Figure Standard and Poor s Inde with dividends reinvested (credit "bull": modification of work b Praitno Hadinata; credit "graph": modification of work b MeasuringWorth)
More information1.7 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 informationSome 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 informationSecond-Order Linear Differential Equations C 2
C8 APPENDIX C Additional Topics in Differential Equations APPENDIX C. Second-Order Homogeneous Linear Equations Second-Order Linear Differential Equations Higher-Order Linear Differential Equations Application
More information4.3 Mean-Value Theorem and Monotonicity
.3 Mean-Value Theorem and Monotonicit 1. Mean Value Theorem Theorem: Suppose that f is continuous on the interval a, b and differentiable on the interval a, b. Then there eists a number c in a, b such
More informationMath 3201 UNIT 5: Polynomial Functions NOTES. Characteristics of Graphs and Equations of Polynomials Functions
1 Math 301 UNIT 5: Polnomial Functions NOTES Section 5.1 and 5.: Characteristics of Graphs and Equations of Polnomials Functions What is a polnomial function? Polnomial Function: - A function that contains
More informationUNCORRECTED 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 informationES.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 informationLESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II
1 LESSON #4 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The
More informationSystems 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 informationLESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II
1 LESSON #8 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The
More informationUnit 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 information6.4 graphs OF logarithmic FUnCTIOnS
SECTION 6. graphs of logarithmic functions 9 9 learning ObjeCTIveS In this section, ou will: Identif the domain of a logarithmic function. Graph logarithmic functions. 6. graphs OF logarithmic FUnCTIOnS
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures AB = BA = I,
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 7 MATRICES II Inverse of a matri Sstems of linear equations Solution of sets of linear equations elimination methods 4
More informationUnit 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 informationLinear 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 informationChapter 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 informationMath 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 informationReview 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 informationMathematics 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 information8.4. If we let x denote the number of gallons pumped, then the price y in dollars can $ $1.70 $ $1.70 $ $1.70 $ $1.
8.4 An Introduction to Functions: Linear Functions, Applications, and Models We often describe one quantit in terms of another; for eample, the growth of a plant is related to the amount of light it receives,
More information2.1 Rates of Change and Limits AP Calculus
.1 Rates of Change and Limits AP Calculus.1 RATES OF CHANGE AND LIMITS Limits Limits are what separate Calculus from pre calculus. Using a it is also the foundational principle behind the two most important
More informationSystems of Linear and Quadratic Equations. Check Skills You ll Need. y x. Solve by Graphing. Solve the following system by graphing.
NY- Learning Standards for Mathematics A.A. Solve a sstem of one linear and one quadratic equation in two variables, where onl factoring is required. A.G.9 Solve sstems of linear and quadratic equations
More informationChapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs
Ch 5 Alg Note Sheet Ke Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs Definition: Standard Form of a Quadratic Function The
More information1.2 Functions and Their Properties PreCalculus
1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given
More informationPolynomial and Rational Functions
Name Date Chapter Polnomial and Rational Functions Section.1 Quadratic Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Important Vocabular Define
More informationStudy Guide and Intervention
6- NAME DATE PERID Stud Guide and Intervention Graphing Quadratic Functions Graph Quadratic Functions Quadratic Function A function defined b an equation of the form f () a b c, where a 0 b Graph of a
More informationIntroduction 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 informationLESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II
LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,
More informationHandout 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 informationDeterminants. We said in Section 3.3 that a 2 2 matrix a b. Determinant of an n n Matrix
3.6 Determinants We said in Section 3.3 that a 2 2 matri a b is invertible if and onl if its c d erminant, ad bc, is nonzero, and we saw the erminant used in the formula for the inverse of a 2 2 matri.
More informationMethods for Advanced Mathematics (C3) Coursework Numerical Methods
Woodhouse College 0 Page Introduction... 3 Terminolog... 3 Activit... 4 Wh use numerical methods?... Change of sign... Activit... 6 Interval Bisection... 7 Decimal Search... 8 Coursework Requirements on
More informationSection 3.1 Solving Linear Systems by Graphing
Section 3.1 Solving Linear Sstems b Graphing Name: Period: Objective(s): Solve a sstem of linear equations in two variables using graphing. Essential Question: Eplain how to tell from a graph of a sstem
More informationChapter 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 informationCh 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 informationf(x) = 2x 2 + 2x - 4
4-1 Graphing Quadratic Functions What You ll Learn Scan the tet under the Now heading. List two things ou will learn about in the lesson. 1. Active Vocabular 2. New Vocabular Label each bo with the terms
More information1.2 Functions and Their Properties PreCalculus
1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given
More informationLimits. Calculus Module C06. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.
e Calculus Module C Limits Copright This publication The Northern Alberta Institute of Technolog. All Rights Reserved. LAST REVISED March, Introduction to Limits Statement of Prerequisite Skills Complete
More informationDemonstrate 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 information3.1 Exponential Functions and Their Graphs
.1 Eponential Functions and Their Graphs Sllabus Objective: 9.1 The student will sketch the graph of a eponential, logistic, or logarithmic function. 9. The student will evaluate eponential or logarithmic
More informationSection 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 informationDerivatives 2: The Derivative at a Point
Derivatives 2: The Derivative at a Point 69 Derivatives 2: The Derivative at a Point Model 1: Review of Velocit In the previous activit we eplored position functions (distance versus time) and learned
More information11.4 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 informationFinding Limits Graphically and Numerically. An Introduction to Limits
8 CHAPTER Limits and Their Properties Section Finding Limits Graphicall and Numericall Estimate a it using a numerical or graphical approach Learn different was that a it can fail to eist Stud and use
More informationReview: critical point or equivalently f a,
Review: a b f f a b f a b critical point or equivalentl f a, b A point, is called a of if,, 0 A local ma or local min must be a critical point (but not conversel) 0 D iscriminant (or Hessian) f f D f f
More informationPolynomial and Rational Functions
Polnomial and Rational Functions Figure -mm film, once the standard for capturing photographic images, has been made largel obsolete b digital photograph. (credit film : modification of work b Horia Varlan;
More informationUnit 10 - Graphing Quadratic Functions
Unit - Graphing Quadratic Functions PREREQUISITE SKILLS: students should be able to add, subtract and multipl polnomials students should be able to factor polnomials students should be able to identif
More informationRoberto s Notes on Integral Calculus Chapter 3: Basics of differential equations Section 3. Separable ODE s
Roberto s Notes on Integral Calculus Chapter 3: Basics of differential equations Section 3 Separable ODE s What ou need to know alread: What an ODE is and how to solve an eponential ODE. What ou can learn
More informationExponential and Logarithmic Functions
Eponential and Logarithmic Functions 6 Figure Electron micrograph of E. Coli bacteria (credit: Mattosaurus, Wikimedia Commons) CHAPTER OUTLINE 6. Eponential Functions 6. Logarithmic Properties 6. Graphs
More informationReview of Essential Skills and Knowledge
Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope
More informationLimits 4: Continuity
Limits 4: Continuit 55 Limits 4: Continuit Model : Continuit I. II. III. IV. z V. VI. z a VII. VIII. IX. Construct Your Understanding Questions (to do in class). Which is the correct value of f (a) in
More informationSolving Linear Systems
1.4 Solving Linear Sstems Essential Question How can ou determine the number of solutions of a linear sstem? A linear sstem is consistent when it has at least one solution. A linear sstem is inconsistent
More informationPre-AP Algebra 2 Lesson 1-1 Basics of Functions
Lesson 1-1 Basics of Functions Objectives: The students will be able to represent functions verball, numericall, smbolicall, and graphicall. The students will be able to determine if a relation is a function
More informationand y f ( x ). given the graph of y f ( x ).
FUNCTIONS AND RELATIONS CHAPTER OBJECTIVES:. Concept of function f : f ( ) : domain, range; image (value). Odd and even functions Composite functions f g; Identit function. One-to-one and man-to-one functions.
More informationGet 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 informationEigenvalues for Triangular Matrices. ENGI 7825: Linear Algebra Review Finding Eigenvalues and Diagonalization
Eigenvalues for Triangular Matrices ENGI 78: Linear Algebra Review Finding Eigenvalues and Diagonalization Adapted from Notes Developed by Martin Scharlemann June 7, 04 The eigenvalues for a triangular
More informationMathematics 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 informationLU 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 informationMA123, Chapter 1: Equations, functions and graphs (pp. 1-15)
MA123, Chapter 1: Equations, functions and graphs (pp. 1-15) Date: Chapter Goals: Identif solutions to an equation. Solve an equation for one variable in terms of another. What is a function? Understand
More information15.4 Equation of a Circle
Name Class Date 1.4 Equation of a Circle Essential Question: How can ou write the equation of a circle if ou know its radius and the coordinates of its center? Eplore G.1.E Show the equation of a circle
More informationName Class Date. Solving Special Systems by Graphing. Does this linear system have a solution? Use the graph to explain.
Name Class Date 5 Solving Special Sstems Going Deeper Essential question: How do ou solve sstems with no or infinitel man solutions? 1 A-REI.3.6 EXAMPLE Solving Special Sstems b Graphing Use the graph
More information14.1 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 informationAlgebra II Notes Unit Six: Polynomials Syllabus Objectives: 6.2 The student will simplify polynomial expressions.
Algebra II Notes Unit Si: Polnomials Sllabus Objectives: 6. The student will simplif polnomial epressions. Review: Properties of Eponents (Allow students to come up with these on their own.) Let a and
More informationChapter 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 informationPre-Calculus Notes Section 12.2 Evaluating Limits DAY ONE: Lets look at finding the following limits using the calculator and algebraically.
Pre-Calculus Notes Name Section. Evaluating Limits DAY ONE: Lets look at finding the following its using the calculator and algebraicall. 4 E. ) 4 QUESTION: As the values get closer to 4, what are the
More informationThe Coordinate Plane and Linear Equations Algebra 1
Name: The Coordinate Plane and Linear Equations Algebra Date: We use the Cartesian Coordinate plane to locate points in two-dimensional space. We can do this b measuring the directed distances the point
More informationf 0 ab a b: base f
Precalculus Notes: Unit Eponential and Logarithmic Functions Sllabus Objective: 9. The student will sketch the graph of a eponential, logistic, or logarithmic function. 9. The student will evaluate eponential
More informationKEY IDEAS. Chapter 1 Function Transformations. 1.1 Horizontal and Vertical Translations Pre-Calculus 12 Student Workbook MHR 1
Chapter Function Transformations. Horizontal and Vertical Translations A translation can move the graph of a function up or down (vertical translation) and right or left (horizontal translation). A translation
More informationSection 4.1 Increasing and Decreasing Functions
Section.1 Increasing and Decreasing Functions The graph of the quadratic function f 1 is a parabola. If we imagine a particle moving along this parabola from left to right, we can see that, while the -coordinates
More informationQuadratic Functions. The graph of the function shifts right 3. The graph of the function shifts left 3.
Quadratic Functions The translation o a unction is simpl the shiting o a unction. In this section, or the most part, we will be graphing various unctions b means o shiting the parent unction. We will go
More information3 Polynomial and Rational Functions
3 Polnomial and Rational Functions 3.1 Quadratic Functions and Models 3.2 Polnomial Functions and Their Graphs 3.3 Dividing Polnomials 3.4 Real Zeros of Polnomials 3.5 Comple Zeros and the Fundamental
More informationCubic and quartic functions
3 Cubic and quartic functions 3A Epanding 3B Long division of polnomials 3C Polnomial values 3D The remainder and factor theorems 3E Factorising polnomials 3F Sum and difference of two cubes 3G Solving
More information6. 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 informationAdditional Topics in Differential Equations
0537_cop6.qd 0/8/08 :6 PM Page 3 6 Additional Topics in Differential Equations In Chapter 6, ou studied differential equations. In this chapter, ou will learn additional techniques for solving differential
More informationAnswer Explanations. The SAT Subject Tests. Mathematics Level 1 & 2 TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE
The SAT Subject Tests Answer Eplanations TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE Mathematics Level & Visit sat.org/stpractice to get more practice and stud tips for the Subject Test
More informationH.Algebra 2 Summer Review Packet
H.Algebra Summer Review Packet 1 Correlation of Algebra Summer Packet with Algebra 1 Objectives A. Simplifing Polnomial Epressions Objectives: The student will be able to: Use the commutative, associative,
More information12.1 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 information4 The Cartesian Coordinate System- Pictures of Equations
The Cartesian Coordinate Sstem- Pictures of Equations Concepts: The Cartesian Coordinate Sstem Graphs of Equations in Two Variables -intercepts and -intercepts Distance in Two Dimensions and the Pthagorean
More informationChapter 8 Notes SN AA U2C8
Chapter 8 Notes SN AA U2C8 Name Period Section 8-: Eploring Eponential Models Section 8-2: Properties of Eponential Functions In Chapter 7, we used properties of eponents to determine roots and some of
More information20.2 Connecting Intercepts and Linear Factors
Name Class Date 20.2 Connecting Intercepts and Linear Factors Essential Question: How are -intercepts of a quadratic function and its linear factors related? Resource Locker Eplore Connecting Factors and
More informationEngineering Mathematics I
Engineering Mathematics I_ 017 Engineering Mathematics I 1. Introduction to Differential Equations Dr. Rami Zakaria Terminolog Differential Equation Ordinar Differential Equations Partial Differential
More informationA function from a set D to a set R is a rule that assigns a unique element in R to each element in D.
1.2 Functions and Their Properties PreCalculus 1.2 FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1.2 1. Determine whether a set of numbers or a graph is a function 2. Find the domain of a function
More informationMathematics. Mathematics 2. hsn.uk.net. Higher HSN22000
hsn.uk.net Higher Mathematics UNIT Mathematics HSN000 This document was produced speciall for the HSN.uk.net website, and we require that an copies or derivative works attribute the work to Higher Still
More information= x. Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions. Math Background
Algebra II Notes Quadratic Functions Unit 3.1 3. Graphing Quadratic Functions Math Background Previousl, ou Identified and graphed linear functions Applied transformations to parent functions Graphed quadratic
More informationk is a product of elementary matrices.
Mathematics, Spring Lecture (Wilson) Final Eam May, ANSWERS Problem (5 points) (a) There are three kinds of elementary row operations and associated elementary matrices. Describe what each kind of operation
More information5. Perform the indicated operation and simplify each of the following expressions:
Precalculus Worksheet.5 1. What is - 1? Just because we refer to solutions as imaginar does not mean that the solutions are meaningless. Fields such as quantum mechanics and electromagnetism depend on
More informationCh 5 Alg 2 L2 Note Sheet Key Do Activity 1 on your Ch 5 Activity Sheet.
Ch Alg L Note Sheet Ke Do Activit 1 on our Ch Activit Sheet. Chapter : Quadratic Equations and Functions.1 Modeling Data With Quadratic Functions You had three forms for linear equations, ou will have
More information6.3 Interpreting Vertex Form and Standard Form
Name Class Date 6.3 Interpreting Verte Form and Standard Form Essential Question: How can ou change the verte form of a quadratic function to standard form? Resource Locker Eplore Identifing Quadratic
More informationIn this chapter a student has to learn the Concept of adjoint of a matrix. Inverse of a matrix. Rank of a matrix and methods finding these.
MATRICES UNIT STRUCTURE.0 Objectives. Introduction. Definitions. Illustrative eamples.4 Rank of matri.5 Canonical form or Normal form.6 Normal form PAQ.7 Let Us Sum Up.8 Unit End Eercise.0 OBJECTIVES In
More informationx. 4. 2x 10 4x. 10 x
CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 1 of Reasoning with Equations and Quantities Name: Date: Understand solving equations as a process of reasoning and eplain the reasoning MCC9-1.A.REI.1 Eplain
More informationChapter 5. Eigenvalues and Eigenvectors
Chapter 5 Eigenvalues and Eigenvectors Section 5. Eigenvectors and Eigenvalues Motivation: Difference equations A Biology Question How to predict a population of rabbits with given dynamics:. half of the
More informationSOLVING SYSTEMS OF EQUATIONS
SOLVING SYSTEMS OF EQUATIONS 4.. 4..4 Students have been solving equations even before Algebra. Now the focus on what a solution means, both algebraicall and graphicall. B understanding the nature of solutions,
More informationMath 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 informationFinding Limits Graphically and Numerically. An Introduction to Limits
60_00.qd //0 :05 PM Page 8 8 CHAPTER Limits and Their Properties Section. Finding Limits Graphicall and Numericall Estimate a it using a numerical or graphical approach. Learn different was that a it can
More informationREVISION 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 information9.12 Quadratics Review
Algebra Name _ B2g0gD6L jkwudtaaa msvopfwtowiarneq CLOLXCa.I K `Awljla `rtiugohhtfs_ QrIefsfeYrZvtetdf. 9.2 Quadratics Review ) What is the difference between the two mathematical statements below? Then
More informationAnalytic Geometry in Three Dimensions
Analtic Geometr in Three Dimensions. The Three-Dimensional Coordinate Sstem. Vectors in Space. The Cross Product of Two Vectors. Lines and Planes in Space The three-dimensional coordinate sstem is used
More information