Systems of Algebraic Equations and Systems of Differential Equations


 Paulina Robinson
 1 years ago
 Views:
Transcription
1 Systems of Algebraic Equations and Systems of Differential Equations Topics: 2 by 2 systems of linear equations Matrix expression; Ax = b Solving 2 by 2 homogeneous systems Functions defined on matrices Determinant of a 2 by 2 matrix Trace of a 2 by 2 matrix Eigenvalues/eigenvectors of a 2 by 2 matrix Characteristic equation Systems of autonomous homogeneous with constant coefficients DEs; two dependent variables Case real eigenvalues Revised 2/22/2016
2 Matrices and Systems of Algebraic Equations A 2 2 system of linear equations is easily represented using matrices. We adopt the following matrix notation. For system ax + bx =k 1 cx + dx =k 2 we identify three matrices associated with this pair of equations. a b x1 k1 A, x, k c d x k 2 2 coefficient matrix column of unknowns rightside column ax 1 + bx 2=k1 In place of the system we use the matrix expression Ax = k. cx + dx =k 2 Writing Ax in matrix algebra indicates a product of matrix A times matrix x. The result of this matrix product using dot products of rows of A with the column x is ax 1 + bx2 Ax cx 1 + dx 2 This is a product of 2 2 matrix A time 2 1 matrix x resulting in a 2 1 matrix which is the left side of the linear system.
3 Example: Linear system 2x  3x = 6 x  2x = 5 in matrix form is denoted Ax = k, where A 2 3 x1 6, x x, k 2 5 The Determinant of a 2 by 2 Matrix Definition: The determinant of a 2 2 matrix computed by the expression det(a) = ad bc.. A a b c d is denoted det(a) and is An easy way to recall this computation is from the diagram. We take the product of the diagonal entries and add to it the negative of the product of the reverse diagonal. Examples: 4 1 x det = 83 =11, det =x + 6, det = 44 = 0, λ 2 2 det =(1  λ)(2  λ) + 6 = λ  3λ λ Greek letter lambda; λ.
4 Trace of a 2 by 2 matrix Definition: The trace of a 2 2 matrix the expression tr(a) = a + d. A a b c d is denoted tr(a) and is computed by The trace function computes the sum of the diagonal entries of the matrix. Examples: tr = 5 + (2) = 3, tr = 4 + (4) = A Special Matrix 1 0 I 0 1 The identity matrix is an example of a diagonal matrix; that is, all entries are zero except possibly those on the diagonal. The letter I is reserved to denote the identity matrix. The next example shows the results of applying our matrix functions to the 2 2 identity matrix. Example: det(i) = 1 and tr(i) = 2. Definition: The 2 2 identity matrix is denoted by the letter I and is defined to be The identity matrix plays an important role in matrix multiplication. If A is any 2 by 2 matrix then by applying matrix multiplication we have that AI = IA = A.
5 Homogeneous 2 by 2 Systems of Algebraic Equations A homogeneous system of 2 equations in 2 unknowns has the form ax + bx = 0 cx + dx = 0 There are just two types of solutions to such systems: 0 (i) A single solution x 1 = 0, x 2 = 0 which we will express as the column vector x. 0 (ii) Infinitely many solutions in which one of the unknowns, either x 1 or x 2, can be chosen arbitrarily. The form of the solution of a homogeneous system is determined by properties of the coefficient matrix A a b c d
6 Omitting the proof we state the following for solutions of system which can be expressed as the matrix equation Ax = 0 ax + bx = 0 0 (i) System cx 1 + dx 2= 0 has only the solution x if and only if 0 (ii) System ax + bx = 0 cx + dx = 0 det(a) det a b adbc 0 c d has infinitely many solutions if and only if det(a) det a b adbc = 0 c d ax + bx = 0 cx + dx = 0 0 x 0 The column of all zeros is called the trivial solution. In upcoming work we are only interested in the second case. The next example illustrates this case.
7 Example: Let 2 1 A, then solve Ax = 0. Computing det(a) = 4 4 = 0. This implies 4 2 2x1 x2 0 there are infinitely many solutions. The linear system can be written in the form for x 1 in each equation giving the system. that 1 x1 x2 x 2 x 2 x2 4x 2x 0. In this system we can solve This pair of equations implies Thus we can compute every solution of the system Ax = 0 by choosing x 2 arbitrarily and so a column represents the set of all possible solutions. x x x x Then we call the general solution of Ax = 0. Often the general solution is represented by letting x 2 = r, any arbitrary value, so then. Note that if r = 0 we get the trivial solution, for r = 1 we get, for r = 2 we get, and for r = 8 we get 4 x 8 1 x x 2 x2 2 1 x 2 1. There are infinitely many nontrivial solutions. 1 r x 2 r 1 x 2
8 Eigenvalues and Eigenvectors of 2 2 matrices Matrices are useful in a wide variety of applications. So you might suspect that we can construct functions involving matrices. As with functions you met in calculus matrix functions have a domain (input information) and a range (output information). One of the simplest matrix functions can be expressed as y = f(x) = Ax where A is a 2 2 matrix, x, an input, is a 2 1 column matrix. This function requires that we perform a matrix product A times x in order to produce an output y, which is also a 2 1 column matrix. An important feature of a matrix function y = f(x) = Ax involves determining input columns x 0 so that the output y is a multiple of x. That is, so that we have Ax = λx, where λ is some number (often called a parameter). (We can think of 2 1 column matrices as vectors in a Cartesian plane, so here we want the output vector y to be parallel to the input vector x.)
9 In the previous example we illustrated that if Ax = λx, then for any number r, A(rx) = λ(rx). Hence once we have one column x so that the output of function y = Ax is parallel to the input x there are infinitely many such columns. From our earlier discussion involving 2 2 homogeneous linear equations Ax = 0 with det(a) = 0 we showed there were infinitely many solutions. So it is likely that the two situations are related. We show there is such a relationship and how to determine columns (vectors) so that when they are multiplied by matrix A the result is a column representing a parallel column (vector). For a 2 2 matrix A we seek 2 1 columns x 0 so that Ax = λx. This matrix expression is called the eigenequation. This equation is actually a nonlinear equation since both column x and number λ are unknown. Definition: Let A be a 2 2 matrix. The number λ is called an eigenvalue of matrix A if there exists a 2 1 column x 0 so that Ax = λx. Every 2 1 column x that satisfies the eigenequation is called an eigenvector of A associated with eigenvalue λ. (Note: by definition an eigenvector of A cannot be a column of all zeros.)
10 Here we can rephrase a previous statement using eigen terminology: If x is an eigenvector associated with eigenvalue, then so is rx for any number r 0. How do we compute eigenvalues and eigenvectors of a 2 2 matrix? We start with the eigenequation Ax = λx and restructure it into a homogeneous system of 1 0 I w w w 2 equations. We previously defined the 2 2 identity matrix. If is any 2 1 matrix it follows that Iw w. (Matrix multiplication is required.) So we can replace x on the right side of the eigenequation to obtain Ax = λix. Applying some matrix algebra we rearrange this expression as Ax λix = 0. Recall that 0 represents the column system 0 0.) Then factoring we have the homogeneous linear (A λi)x = 0.
11 In our previous discussion of solutions of homogeneous systems, we stated that a homogeneous system had a nontrivial solution (that is, 0 x 0 ) only if the determinant of the coefficient matrix is zero. Since an eigenvector cannot be the zero column it must be that for linear system (A λi)x = 0 to have a nontrivial solution that the determinant of the coefficient matrix is zero; that is, det(a λi) = 0. (Since matrices A and I are known it is the value of eigenvalue λ that controls when the determinant is zero.) Definition: The determinant det(a λi) is called the characteristic polynomial of A and the equation det(a λi) = 0 is called the characteristic equation of A. Thus we have a compact expression for the characteristic equation in terms of the trace and determinant of matrix A. det(a λi) = λ 2 tr(a)λ + det(a) = 0
12 For the 2 2 matrix A the characteristic equation is a quadratic and its roots are the eigenvalues of A. Thus we know there are exactly two eigenvalues for 2 2 matrix A. Once we have computed the eigenvalues we return to the homogeneous linear system (A λi)x = 0 to find an associated eigenvector for each of the eigenvalues. We illustrate this in the next example.
13
14 Continuing with Step 2.
15 Summary: For our eigenvalue/eigenvector computations we use the following twostep procedure. Step 1. Determine the characteristic equation is λ 2 tr(a)λ + det(a) = 0 and solve for the eigenvalues. Step 2. For each eigenvalue λ determine a nontrivial solution to the homogeneous linear system (A λi)x = 0. The nontrivial solution will provide the set of eigenvectors associated with the eigenvalue λ.
16 For 2 2 matrices the characteristic equation is a quadratic so there are three types of roots, hence eigenvalues. They are real and distinct, λ 1 λ 2, real but repeated, λ 1 = λ 2, and complex λ ± µi, µ 0. In the preceding example that matrix had real distinct eigenvalues λ 1 = 4 and λ 2 = 1. A 2 2 matrix A with real entries can have complex eigenvalues. In such a case the corresponding eigenvectors also have complex entries. For now we focus on the case where the eigenvalues are real and distinct.
17 Solving systems of two autonomous homogeneous linear differential equations with constant coefficients A system of two autonomous homogeneous linear differential equations with constant coefficients in independent variable t and dependent variables u 1 (t) and u 2 (t) has the form u '(t) = a u (t) + a u (t) u '(t) = a u (t) + a u (t) Often the inclusion of the independent variable t is omitted to provide a more compact expression, then the system appears in the form u ' = a u + a u u ' = a u + a u
18 Our goal is to show how the eigenvalues and eigenvectors play a role in determining the general solution, the set of all solutions, to this system of differential equations. The proof of the following solution procedure requires some matrix theory beyond the scope of this introduction. But it is reminiscent of the technique for solving a homogeneous second order linear DE with constant coefficients. Procedure for obtaining the general solution to u' = Au when the eigenvalues of the a c b d 2 2 coefficient matrix A are real and distinct λ 1 λ 2. Step 1. Find the eigenvalues of matrix A. Call them λ 1 and λ 2. 2 Use the quadratic tr(a) det(a) 0 find the roots. Step 2. Compute the associated eigenvectors for each eigenvalue. Call them v and w. To find v compute a nonzero solution to the system a λ v + bv = 0 1 c v + d λ v = 0 1 similarly for w find a nonzero solution to Step 3. The general solution of u' = Au is given by the equation u 1(t) 1t 2t u(t) C1e v C2e w u 2(t) where C 1 and C 2 are arbitrary constants. a λ w + b w = 0 2 Note the exponential functions that appear here. c w + d λ w = 0 2
19 We illustrate the use of this procedure with examples. Example: Let s return to our example involving matrix A where now we have the system of differential equations u 1'(t) 2 3u 1(t) u'. Previously we u 2 '(t) Αυ 2 1 u2( t) computed the eigen information for this coefficient matrix as Using the expression for the general solution u (t) t t u(t) C e v C e v u (t) we get Now let s consider initial conditions for this system of differential equations so we can determine the arbitrary constants.
20
21 Another example: Determine the solution of the IVP given by u '(t)  141u (t)  54u (t) 1 u '(t) 360 u (t) 138u (t) 2 The coefficient matrix is A= tr(a) = 3 and det(a) = So the characteristic polynomial is The roots are the eigenvalues : λ 1= 3, λ 2 = λ  tr(a)λ +det(a) = λ  (3)λ  18 = λ +3λ  18 To find the eigenvectors we solve the homogeneous linear systems (A  λ 1 x) = 0 and (A λ 2 x) = 0. We find the eigenvectors which can be chosen to have integer entries are respectively Then the solution of the system of DEs is 32 v 1 = and v 2 = 8 5 u1() t 3t 3 6t 2 u(t) C1e C2e u2() t 8 5
22 If we had initial conditions u 1 (0) = 2 and u 2 (0) = 1, then we must solve the system C We find C 1 = 12 and C 2 = 19. The solution of the IVP C u '(t) = u (t)  54u (t) 1 u '(t) = 360 u (t) + 138u (t) 2 u 1(0) = 2, u 2(0) = 1 u () t 3 2 u(t) 12e 19e u2() t 8 5 is 1 3t 6t
A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More informationLecture Notes: Eigenvalues and Eigenvectors. 1 Definitions. 2 Finding All Eigenvalues
Lecture Notes: Eigenvalues and Eigenvectors Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk 1 Definitions Let A be an n n matrix. If there
More informationEcon Slides from Lecture 7
Econ 205 Sobel Econ 205  Slides from Lecture 7 Joel Sobel August 31, 2010 Linear Algebra: Main Theory A linear combination of a collection of vectors {x 1,..., x k } is a vector of the form k λ ix i for
More informationand 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 informationav 1 x 2 + 4y 2 + xy + 4z 2 = 16.
74 85 Eigenanalysis The subject of eigenanalysis seeks to find a coordinate system, in which the solution to an applied problem has a simple expression Therefore, eigenanalysis might be called the method
More informationLinear Algebra Primer
Linear Algebra Primer David Doria daviddoria@gmail.com Wednesday 3 rd December, 2008 Contents Why is it called Linear Algebra? 4 2 What is a Matrix? 4 2. Input and Output.....................................
More informationCalculating determinants for larger matrices
Day 26 Calculating determinants for larger matrices We now proceed to define det A for n n matrices A As before, we are looking for a function of A that satisfies the product formula det(ab) = det A det
More informationA matrix times a column vector is the linear combination of the columns of the matrix weighted by the entries in the column vector.
18.03 Class 33, Apr 28, 2010 Eigenvalues and eigenvectors 1. Linear algebra 2. Ray solutions 3. Eigenvalues 4. Eigenvectors 5. Initial values [1] Prologue on Linear Algebra. Recall [a b ; c d] [x ; y]
More informationLS.2 Homogeneous Linear Systems with Constant Coefficients
LS2 Homogeneous Linear Systems with Constant Coefficients Using matrices to solve linear systems The naive way to solve a linear system of ODE s with constant coefficients is by eliminating variables,
More information9.1 Eigenanalysis I Eigenanalysis II Advanced Topics in Linear Algebra Kepler s laws
Chapter 9 Eigenanalysis Contents 9. Eigenanalysis I.................. 49 9.2 Eigenanalysis II................. 5 9.3 Advanced Topics in Linear Algebra..... 522 9.4 Kepler s laws................... 537
More information6 EIGENVALUES AND EIGENVECTORS
6 EIGENVALUES AND EIGENVECTORS INTRODUCTION TO EIGENVALUES 61 Linear equations Ax = b come from steady state problems Eigenvalues have their greatest importance in dynamic problems The solution of du/dt
More informationEIGENVALUES AND EIGENVECTORS
EIGENVALUES AND EIGENVECTORS Diagonalizable linear transformations and matrices Recall, a matrix, D, is diagonal if it is square and the only nonzero entries are on the diagonal This is equivalent to
More information4. Determinants.
4. Determinants 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 2 2 determinant 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 3 3 determinant 4.1.
More informationIntroduction to Matrix Algebra
Introduction to Matrix Algebra August 18, 2010 1 Vectors 1.1 Notations A pdimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the line. When p
More informationEigenvalues and Eigenvectors. Review: Invertibility. Eigenvalues and Eigenvectors. The Finite Dimensional Case. January 18, 2018
January 18, 2018 Contents 1 2 3 4 Review 1 We looked at general determinant functions proved that they are all multiples of a special one, called det f (A) = f (I n ) det A. Review 1 We looked at general
More informationMatrices and Deformation
ES 111 Mathematical Methods in the Earth Sciences Matrices and Deformation Lecture Outline 13  Thurs 9th Nov 2017 Strain Ellipse and Eigenvectors One way of thinking about a matrix is that it operates
More information0.1 Eigenvalues and Eigenvectors
0.. EIGENVALUES AND EIGENVECTORS MATH 22AL Computer LAB for Linear Algebra Eigenvalues and Eigenvectors Dr. Daddel Please save your MATLAB Session (diary)as LAB9.text and submit. 0. Eigenvalues and Eigenvectors
More informationc c c c c c c c c c a 3x3 matrix C= has a determinant determined by
Linear Algebra Determinants and Eigenvalues Introduction: Many important geometric and algebraic properties of square matrices are associated with a single real number revealed by what s known as the determinant.
More informationMath 315: Linear Algebra Solutions to Assignment 7
Math 5: Linear Algebra s to Assignment 7 # Find the eigenvalues of the following matrices. (a.) 4 0 0 0 (b.) 0 0 9 5 4. (a.) The characteristic polynomial det(λi A) = (λ )(λ )(λ ), so the eigenvalues are
More informationMATHEMATICS FOR ENGINEERS BASIC MATRIX THEORY TUTORIAL 3  EIGENVECTORS AND EIGENVALUES
MATHEMATICS FOR ENGINEERS BASIC MATRIX THEORY TUTORIAL 3  EIGENVECTORS AND EIGENVALUES This is the third tutorial on matrix theory. It is entirely devoted to the subject of Eigenvectors and Eigenvalues
More informationChapters 5 & 6: Theory Review: Solutions Math 308 F Spring 2015
Chapters 5 & 6: Theory Review: Solutions Math 308 F Spring 205. If A is a 3 3 triangular matrix, explain why det(a) is equal to the product of entries on the diagonal. If A is a lower triangular or diagonal
More informationLinear algebra II Homework #1 solutions A = This means that every eigenvector with eigenvalue λ = 1 must have the form
Linear algebra II Homework # solutions. Find the eigenvalues and the eigenvectors of the matrix 4 6 A =. 5 Since tra = 9 and deta = = 8, the characteristic polynomial is f(λ) = λ (tra)λ+deta = λ 9λ+8 =
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More informationEigenvalues and Eigenvectors
LECTURE 3 Eigenvalues and Eigenvectors Definition 3.. Let A be an n n matrix. The eigenvalueeigenvector problem for A is the problem of finding numbers λ and vectors v R 3 such that Av = λv. If λ, v are
More informationEigenvalues and Eigenvectors
CHAPTER Eigenvalues and Eigenvectors CHAPTER CONTENTS. Eigenvalues and Eigenvectors 9. Diagonalization. Complex Vector Spaces.4 Differential Equations 6. Dynamical Systems and Markov Chains INTRODUCTION
More informationCh 10.1: Two Point Boundary Value Problems
Ch 10.1: Two Point Boundary Value Problems In many important physical problems there are two or more independent variables, so the corresponding mathematical models involve partial differential equations.
More informationLinear Algebra Final Exam Review
Linear Algebra Final Exam Review. Let A be invertible. Show that, if v, v, v 3 are linearly independent vectors, so are Av, Av, Av 3. NOTE: It should be clear from your answer that you know the definition.
More informationMAT 1302B Mathematical Methods II
MAT 1302B Mathematical Methods II Alistair Savage Mathematics and Statistics University of Ottawa Winter 2015 Lecture 19 Alistair Savage (uottawa) MAT 1302B Mathematical Methods II Winter 2015 Lecture
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More informationEigenvalues and Eigenvectors
Eigenvalues and Eigenvectors Philippe B. Laval KSU Fall 2015 Philippe B. Laval (KSU) Eigenvalues and Eigenvectors Fall 2015 1 / 14 Introduction We define eigenvalues and eigenvectors. We discuss how to
More informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationMath 2331 Linear Algebra
5. Eigenvectors & Eigenvalues Math 233 Linear Algebra 5. Eigenvectors & Eigenvalues ShangHuan Chiu Department of Mathematics, University of Houston schiu@math.uh.edu math.uh.edu/ schiu/ ShangHuan Chiu,
More informationIntroduction to Matrices
POLS 704 Introduction to Matrices Introduction to Matrices. The Cast of Characters A matrix is a rectangular array (i.e., a table) of numbers. For example, 2 3 X 4 5 6 (4 3) 7 8 9 0 0 0 Thismatrix,with4rowsand3columns,isoforder
More informationDeterminants by Cofactor Expansion (III)
Determinants by Cofactor Expansion (III) Comment: (Reminder) If A is an n n matrix, then the determinant of A can be computed as a cofactor expansion along the jth column det(a) = a1j C1j + a2j C2j +...
More information1 Linear Regression and Correlation
Math 10B with Professor Stankova Worksheet, Discussion #27; Tuesday, 5/1/2018 GSI name: Roy Zhao 1 Linear Regression and Correlation 1.1 Concepts 1. Often when given data points, we want to find the line
More informationLinear vector spaces and subspaces.
Math 2051 W2008 Margo Kondratieva Week 1 Linear vector spaces and subspaces. Section 1.1 The notion of a linear vector space. For the purpose of these notes we regard (m 1)matrices as mdimensional vectors,
More informationMATH 369 Linear Algebra
Assignment # Problem # A father and his two sons are together 00 years old. The father is twice as old as his older son and 30 years older than his younger son. How old is each person? Problem # 2 Determine
More informationSummer Session Practice Final Exam
Math 2F Summer Session 25 Practice Final Exam Time Limit: Hours Name (Print): Teaching Assistant This exam contains pages (including this cover page) and 9 problems. Check to see if any pages are missing.
More informationMath 215 HW #9 Solutions
Math 5 HW #9 Solutions. Problem 4.4.. If A is a 5 by 5 matrix with all a ij, then det A. Volumes or the big formula or pivots should give some upper bound on the determinant. Answer: Let v i be the ith
More informationHOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION)
HOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION) PROFESSOR STEVEN MILLER: BROWN UNIVERSITY: SPRING 2007 1. CHAPTER 1: MATRICES AND GAUSSIAN ELIMINATION Page 9, # 3: Describe
More informationChapter 2:Determinants. Section 2.1: Determinants by cofactor expansion
Chapter 2:Determinants Section 2.1: Determinants by cofactor expansion [ ] a b Recall: The 2 2 matrix is invertible if ad bc 0. The c d ([ ]) a b function f = ad bc is called the determinant and it associates
More informationChapter Two Elements of Linear Algebra
Chapter Two Elements of Linear Algebra Previously, in chapter one, we have considered single first order differential equations involving a single unknown function. In the next chapter we will begin to
More informationEigenvalues and Eigenvectors
Eigenvalues and Eigenvectors week 2 Fall 26 Eigenvalues and eigenvectors The most simple linear transformation from R n to R n may be the transformation of the form: T (x,,, x n ) (λ x, λ 2,, λ n x n
More informationLinear Algebra: Sample Questions for Exam 2
Linear Algebra: Sample Questions for Exam 2 Instructions: This is not a comprehensive review: there are concepts you need to know that are not included. Be sure you study all the sections of the book and
More informationChapter 1: Systems of Linear Equations and Matrices
: Systems of Linear Equations and Matrices Multiple Choice Questions. Which of the following equations is linear? (A) x + 3x 3 + 4x 4 3 = 5 (B) 3x x + x 3 = 5 (C) 5x + 5 x x 3 = x + cos (x ) + 4x 3 = 7.
More informationMA 265 FINAL EXAM Fall 2012
MA 265 FINAL EXAM Fall 22 NAME: INSTRUCTOR S NAME:. There are a total of 25 problems. You should show work on the exam sheet, and pencil in the correct answer on the scantron. 2. No books, notes, or calculators
More informationQuadratic forms. Here. Thus symmetric matrices are diagonalizable, and the diagonalization can be performed by means of an orthogonal matrix.
Quadratic forms 1. Symmetric matrices An n n matrix (a ij ) n ij=1 with entries on R is called symmetric if A T, that is, if a ij = a ji for all 1 i, j n. We denote by S n (R) the set of all n n symmetric
More informationLinear algebra and applications to graphs Part 1
Linear algebra and applications to graphs Part 1 Written up by Mikhail Belkin and Moon Duchin Instructor: Laszlo Babai June 17, 2001 1 Basic Linear Algebra Exercise 1.1 Let V and W be linear subspaces
More information1. General Vector Spaces
1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule
More informationMATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)
MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m
More informationMath 240 Calculus III
Generalized Calculus III Summer 2015, Session II Thursday, July 23, 2015 Agenda 1. 2. 3. 4. Motivation Defective matrices cannot be diagonalized because they do not possess enough eigenvectors to make
More information18.S34 linear algebra problems (2007)
18.S34 linear algebra problems (2007) Useful ideas for evaluating determinants 1. Row reduction, expanding by minors, or combinations thereof; sometimes these are useful in combination with an induction
More informationLinear Algebra Practice Problems
Linear Algebra Practice Problems Math 24 Calculus III Summer 25, Session II. Determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless otherwise stated,
More informationIr O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v )
Section 3.2 Theorem 3.6. Let A be an m n matrix of rank r. Then r m, r n, and, by means of a finite number of elementary row and column operations, A can be transformed into the matrix ( ) Ir O D = 1 O
More information1. Select the unique answer (choice) for each problem. Write only the answer.
MATH 5 Practice Problem Set Spring 7. Select the unique answer (choice) for each problem. Write only the answer. () Determine all the values of a for which the system has infinitely many solutions: x +
More informationMATH 300, Second Exam REVIEW SOLUTIONS. NOTE: You may use a calculator for this exam You only need something that will perform basic arithmetic.
MATH 300, Second Exam REVIEW SOLUTIONS NOTE: You may use a calculator for this exam You only need something that will perform basic arithmetic. [ ] [ ] 2 2. Let u = and v =, Let S be the parallelegram
More informationMA22S3 Summary Sheet: Ordinary Differential Equations
MA22S3 Summary Sheet: Ordinary Differential Equations December 14, 2017 Kreyszig s textbook is a suitable guide for this part of the module. Contents 1 Terminology 1 2 First order separable 2 2.1 Separable
More information7 Planar systems of linear ODE
7 Planar systems of linear ODE Here I restrict my attention to a very special class of autonomous ODE: linear ODE with constant coefficients This is arguably the only class of ODE for which explicit solution
More informationExtra Problems for Math 2050 Linear Algebra I
Extra Problems for Math 5 Linear Algebra I Find the vector AB and illustrate with a picture if A = (,) and B = (,4) Find B, given A = (,4) and [ AB = A = (,4) and [ AB = 8 If possible, express x = 7 as
More informationMath Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88
Math Camp 2010 Lecture 4: Linear Algebra Xiao Yu Wang MIT Aug 2010 Xiao Yu Wang (MIT) Math Camp 2010 08/10 1 / 88 Linear Algebra Game Plan Vector Spaces Linear Transformations and Matrices Determinant
More information1 Matrices and Systems of Linear Equations
March 3, 203 66. Systems of Linear Equations Matrices and Systems of Linear Equations An m n matrix is an array A = a ij of the form a a n a 2 a 2n... a m a mn where each a ij is a real or complex number.
More information1 Positive definiteness and semidefiniteness
Positive definiteness and semidefiniteness Zdeněk Dvořák May 9, 205 For integers a, b, and c, let D(a, b, c) be the diagonal matrix with + for i =,..., a, D i,i = for i = a +,..., a + b,. 0 for i = a +
More informationMath Linear Algebra Final Exam Review Sheet
Math 151 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a componentwise operation. Two vectors v and w may be added together as long as they contain the same number n of
More informationTopics. Vectors (column matrices): Vector addition and scalar multiplication The matrix of a linear function y Ax The elements of a matrix A : A ij
Topics Vectors (column matrices): Vector addition and scalar multiplication The matrix of a linear function y Ax The elements of a matrix A : A ij or a ij lives in row i and column j Definition of a matrix
More informationLINEAR ALGEBRA MICHAEL PENKAVA
LINEAR ALGEBRA MICHAEL PENKAVA 1. Linear Maps Definition 1.1. If V and W are vector spaces over the same field K, then a map λ : V W is called a linear map if it satisfies the two conditions below: (1)
More informationMATRICES. knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns.
MATRICES After studying this chapter you will acquire the skills in knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns. List of
More informationQuick Tour of Linear Algebra and Graph Theory
Quick Tour of Linear Algebra and Graph Theory CS224W: Social and Information Network Analysis Fall 2014 David Hallac Based on Peter Lofgren, Yu Wayne Wu, and Borja Pelato s previous versions Matrices and
More information4. Linear transformations as a vector space 17
4 Linear transformations as a vector space 17 d) 1 2 0 0 1 2 0 0 1 0 0 0 1 2 3 4 32 Let a linear transformation in R 2 be the reflection in the line = x 2 Find its matrix 33 For each linear transformation
More informationMath 240 Calculus III
The Calculus III Summer 2015, Session II Wednesday, July 8, 2015 Agenda 1. of the determinant 2. determinants 3. of determinants What is the determinant? Yesterday: Ax = b has a unique solution when A
More information20D  Homework Assignment 5
Brian Bowers TA for Hui Sun MATH D Homework Assignment 5 November 8, 3 D  Homework Assignment 5 First, I present the list of all matrix row operations. We use combinations of these steps to row reduce
More informationSPRING 2006 PRELIMINARY EXAMINATION SOLUTIONS
SPRING 006 PRELIMINARY EXAMINATION SOLUTIONS 1A. Let G be the subgroup of the free abelian group Z 4 consisting of all integer vectors (x, y, z, w) such that x + 3y + 5z + 7w = 0. (a) Determine a linearly
More informationSOLUTIONS: ASSIGNMENT Use Gaussian elimination to find the determinant of the matrix. = det. = det = 1 ( 2) 3 6 = 36. v 4.
SOLUTIONS: ASSIGNMENT 9 66 Use Gaussian elimination to find the determinant of the matrix det 1 1 4 4 1 1 1 1 8 8 = det = det 0 7 9 0 0 0 6 = 1 ( ) 3 6 = 36 = det = det 0 0 6 1 0 0 0 6 61 Consider a 4
More informationLinear Algebra, part 2 Eigenvalues, eigenvectors and least squares solutions
Linear Algebra, part 2 Eigenvalues, eigenvectors and least squares solutions AnnaKarin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2013 Main problem of linear algebra 2: Given
More information2 b 3 b 4. c c 2 c 3 c 4
OHSx XM511 Linear Algebra: Multiple Choice Questions for Chapter 4 a a 2 a 3 a 4 b b 1. What is the determinant of 2 b 3 b 4 c c 2 c 3 c 4? d d 2 d 3 d 4 (a) abcd (b) abcd(a b)(b c)(c d)(d a) (c) abcd(a
More informationVector Spaces ปร ภ ม เวกเตอร
Vector Spaces ปร ภ ม เวกเตอร 1 5.1 Real Vector Spaces ปร ภ ม เวกเตอร ของจ านวนจร ง Vector Space Axioms (1/2) Let V be an arbitrary nonempty set of objects on which two operations are defined, addition
More informationMATH 304 Linear Algebra Lecture 33: Bases of eigenvectors. Diagonalization.
MATH 304 Linear Algebra Lecture 33: Bases of eigenvectors. Diagonalization. Eigenvalues and eigenvectors of an operator Definition. Let V be a vector space and L : V V be a linear operator. A number λ
More informationA = 3 B = A 1 1 matrix is the same as a number or scalar, 3 = [3].
Appendix : A Very Brief Linear ALgebra Review Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics Very often in this course we study the shapes
More informationEigenvalues, Eigenvectors, and Diagonalization
Math 240 TA: Shuyi Weng Winter 207 February 23, 207 Eigenvalues, Eigenvectors, and Diagonalization The concepts of eigenvalues, eigenvectors, and diagonalization are best studied with examples. We will
More informationLinear Algebra for Machine Learning. Sargur N. Srihari
Linear Algebra for Machine Learning Sargur N. srihari@cedar.buffalo.edu 1 Overview Linear Algebra is based on continuous math rather than discrete math Computer scientists have little experience with it
More informationLinear Algebra. Paul Yiu. 6D: 2planes in R 4. Department of Mathematics Florida Atlantic University. Fall 2011
Linear Algebra Paul Yiu Department of Mathematics Florida Atlantic University Fall 2011 6D: 2planes in R 4 The angle between a vector and a plane The angle between a vector v R n and a subspace V is the
More information18.06 Professor Strang Quiz 3 May 2, 2008
18.06 Professor Strang Quiz 3 May 2, 2008 Your PRINTED name is: Grading 1 2 3 Please circle your recitation: 1) M 2 2131 A. Ritter 2085 21192 afr 2) M 2 4149 A. Tievsky 2492 34093 tievsky 3) M 3
More information(a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? Solution: dim N(A) 1, since rank(a) 3. Ax =
. (5 points) (a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? dim N(A), since rank(a) 3. (b) If we also know that Ax = has no solution, what do we know about the rank of A? C(A)
More informationChapter 1 Vector Spaces
Chapter 1 Vector Spaces PerOlof Persson persson@berkeley.edu Department of Mathematics University of California, Berkeley Math 110 Linear Algebra Vector Spaces Definition A vector space V over a field
More informationFundamentals of Linear Algebra. Marcel B. Finan Arkansas Tech University c All Rights Reserved
Fundamentals of Linear Algebra Marcel B. Finan Arkansas Tech University c All Rights Reserved 2 PREFACE Linear algebra has evolved as a branch of mathematics with wide range of applications to the natural
More informationDifferential equations
Differential equations Math 7 Spring Practice problems for April Exam Problem Use the method of elimination to find the xcomponent of the general solution of x y = 6x 9x + y = x 6y 9y Soln: The system
More informationChapter 7. Linear Algebra: Matrices, Vectors,
Chapter 7. Linear Algebra: Matrices, Vectors, Determinants. Linear Systems Linear algebra includes the theory and application of linear systems of equations, linear transformations, and eigenvalue problems.
More informationMATRIX ALGEBRA. or x = (x 1,..., x n ) R n. y 1 y 2. x 2. x m. y m. y = cos θ 1 = x 1 L x. sin θ 1 = x 2. cos θ 2 = y 1 L y.
as Basics Vectors MATRIX ALGEBRA An array of n real numbers x, x,, x n is called a vector and it is written x = x x n or x = x,, x n R n prime operation=transposing a column to a row Basic vector operations
More informationMTH 102: Linear Algebra Department of Mathematics and Statistics Indian Institute of Technology  Kanpur. Problem Set
MTH 102: Linear Algebra Department of Mathematics and Statistics Indian Institute of Technology  Kanpur Problem Set 6 Problems marked (T) are for discussions in Tutorial sessions. 1. Find the eigenvalues
More informationEigenvalues, Eigenvectors, and Diagonalization
Week12 Eigenvalues, Eigenvectors, and Diagonalization 12.1 Opening Remarks 12.1.1 Predicting the Weather, Again Let us revisit the example from Week 4, in which we had a simple model for predicting the
More informationSolutions to Final Exam 2011 (Total: 100 pts)
Page of 5 Introduction to Linear Algebra November 7, Solutions to Final Exam (Total: pts). Let T : R 3 R 3 be a linear transformation defined by: (5 pts) T (x, x, x 3 ) = (x + 3x + x 3, x x x 3, x + 3x
More informationSystems of Linear Equations. By: Tri Atmojo Kusmayadi and Mardiyana Mathematics Education Sebelas Maret University
Systems of Linear Equations By: Tri Atmojo Kusmayadi and Mardiyana Mathematics Education Sebelas Maret University Standard of Competency: Understanding the properties of systems of linear equations, matrices,
More informationLecture 12: Diagonalization
Lecture : Diagonalization A square matrix D is called diagonal if all but diagonal entries are zero: a a D a n 5 n n. () Diagonal matrices are the simplest matrices that are basically equivalent to vectors
More informationCalculus for the Life Sciences II Assignment 6 solutions. f(x, y) = 3π 3 cos 2x + 2 sin 3y
Calculus for the Life Sciences II Assignment 6 solutions Find the tangent plane to the graph of the function at the point (0, π f(x, y = 3π 3 cos 2x + 2 sin 3y Solution: The tangent plane of f at a point
More informationIntroduction to Matrices
214 Analysis and Design of Feedback Control Systems Introduction to Matrices Derek Rowell October 2002 Modern system dynamics is based upon a matrix representation of the dynamic equations governing the
More informationELEMENTARY LINEAR ALGEBRA WITH APPLICATIONS. 1. Linear Equations and Matrices
ELEMENTARY LINEAR ALGEBRA WITH APPLICATIONS KOLMAN & HILL NOTES BY OTTO MUTZBAUER 11 Systems of Linear Equations 1 Linear Equations and Matrices Numbers in our context are either real numbers or complex
More informationFinal Exam. Linear Algebra Summer 2011 Math S2010X (3) Corrin Clarkson. August 10th, Solutions
Final Exam Linear Algebra Summer Math SX (3) Corrin Clarkson August th, Name: Solutions Instructions: This is a closed book exam. You may not use the textbook, notes or a calculator. You will have 9 minutes
More informationMATH 431: FIRST MIDTERM. Thursday, October 3, 2013.
MATH 431: FIRST MIDTERM Thursday, October 3, 213. (1) An inner product on the space of matrices. Let V be the vector space of 2 2 real matrices (that is, the algebra Mat 2 (R), but without the mulitiplicative
More informationVector Spaces ปร ภ ม เวกเตอร
Vector Spaces ปร ภ ม เวกเตอร 5.1 Real Vector Spaces ปร ภ ม เวกเตอร ของจ านวนจร ง Vector Space Axioms (1/2) Let V be an arbitrary nonempty set of objects on which two operations are defined, addition and
More informationMATH 310, REVIEW SHEET 2
MATH 310, REVIEW SHEET 2 These notes are a very short summary of the key topics in the book (and follow the book pretty closely). You should be familiar with everything on here, but it s not comprehensive,
More informationLinearization of Differential Equation Models
Linearization of Differential Equation Models 1 Motivation We cannot solve most nonlinear models, so we often instead try to get an overall feel for the way the model behaves: we sometimes talk about looking
More information