Linear Algebra MATH 2270
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Date Created: 10/26/15
Mathematics 22702280 Linear AlgebraIntroduction to Differential Equations Sequence Overview and Instructor Notes Coordinator Nick Korevaar7 LCB 2047 581 73187 korevaar mathutahedu 2270 2280 directory http wwwmathutahedu korevaarcoord2270 2280 August 67 2008 1 Sequence overview Mathematics 2270 2280 is a year long sequence devoted to linear mathematics The rst semester is a course about linear algebra and the second semester is an introduction to ordinary and partial differential equations Most students taking this course are math majors or minors and will take the entire sequence Strong engineering and science students are also encouraged to take 227 0 2280 as a more complete alternative to the engineering math courses 22507 3150 Since you should keep both semesters of the sequence in mind even if you are teaching only one of them7 these notes cover the entire year Students in these two courses are expected to learn the theoretical framework of the mathematics being discussed as well as the practical computational methods which result from the theory In particular7 students should be required to learn key de nitions and proofs7 especially in the linear algebra course The text by Bretscher includes truefalse questions in which students must justify their answers7 and you should use these to help guide your class towards conceptual thought I always include a few theoretical questions on my exams7 sort of warmups for what students will be confronted with in higher level math courses For example7 can a student prove that the general solution to an inhomogeneous linear equation is given by the sum of a particular solution with the general solution to the homogeneous equation Can they de ne a linear operator Can they derive the general solution to a rst order linear differential equation and explain what its form has to do with the previous linear algebra problems Of course7 I give students some indication of the de nitions and theorems they are responsible for before I test them Computationally7 it is expected that students will become pro cient in using Maple or equiv alent software on the computer in order to complete large or complicated computations7 and for visualization If you believe strongly in Matlab vs Maple7 you are free to teach your students via that particular package instead It is not assumed that Math 2270 students have previous MapleMatlab experience7 and the rst project should partly be an introduction to the software being used It is not an option to omit computer projects from the coursework l have taught Math 2270 and Math 2280 several times7 and encourage you to use my old projects or modify them as you please You can nd them by following links from my home page If you choose to expand upon my efforts or to create your own7 please let me know about your successful ideas so that we may maintain an on line collection of links 2 Texts 0 Math 2270 Linear Algebra with Applications 3E by O Bretscher lSBN9780131453340 0 Math 2280 Di erential Equations and Boundary Value Problems Computing and Modeling fourth edition by CH Edwards and DE Penney lSBN9780131561076 3 Prerequisites For Math 2270 students need to have succeeded at rst year Calculus Mathematics 1210 1220 or 1250 1260 or 1270 1280 Math 2270 is a prerequisite for Math 2280 The 2280 students would also ben t from the multivariable calculus in either 1260 or 2210 they need an understand ing of curves and tangent vectors to understand the geometric meaning of solutions to systems of differential equations and they should understand partial derivatives and the chain rule to under stand linearization especially near equilibria of non linear systems and to make sense of partial differential equations 4 Grading In each course there should be at least two in class midterms as well as a comprehensive nal exam Exams should be graded primarily by the instructor Homework should be assigned regularly The assigned homework should be collected and at least partially graded or there should be frequent short quizzes on course and homework material It is important that students get frequent timely feedback on their work Contact Angie Gardiner 585 9478 gardinermathutahedu to request a grader for your section Do this ASAP preferably before the semester begins 5 Computer projects The Department strongly suggests that you include computer work in both Math 2270 and Math 2280 Software output enables you to discuss interesting examples in class which would be too dif cult to work by hand As well the computer is great for expressing and visualizing quantitative and qualitative behavior Your students may not know whether their interests are primarily pure or applied and these two gateway courses to the major should include elements which are theoretical as well as ones which are applied My preference is to use MAPLE although instructors in these classes have also used MATLAB successfully Hopefully by the end of this sequence students will feel comfortably using math software to tackle computationally challenging problems or even as an aid to understanding theoretical concepts As the year progresses I often include homework problems which have a mathematical software component in addition to the larger projects For the rst couple of projects in Math 2270 I recommend meeting with your class in the computer lab LCB 115 as there is no assumption that your students have any previous experience with Maple or Matlab Contact Angie Gardiner before the semester starts if possible to reserve the LCB 115 computer lab for the By the end of 2280 I simply post the projects and let the students work on them outside of class I do often hold some of my of ce hours in the Rushing Student Center lab when a project due date is approaching 6 Prerequisites For Math 2270 students need to have succeeded at rst year Calculus Mathematics 1210 1220 or 1250 1260 Math 2270 is a prerequisite for Math 2280 The 2280 students would also ben t from the multivariable calculus in either 1260 or 2210 they need an understanding of curves and tangent vectors to understand the geometric meaning of solutions to systems of differential equations and they should understand partial derivatives and the chain rule to understand the partial differential equations 7 Math 2270 details Covering the rst 8 chapters of the text is de nitely possible although a bit rushed The Spectral Theorem for symmetric matrices and singular value decomposition Chapter 8 are very important concepts so if at all possible complete this chapter This text is excellent at integrating geometric and algebraic concepts It is written at a level of sophistication that is above what students coming out of Calculus are used to however so make sure to slow down and ll in details at dif cult conceptual junctures The semester begins with the basic study of linear equations the algebra and geometry of linear transformations matrix and transformation inverses in Euclidean space I have a project on fractals via interated function systems which I have taught in my own class at this point of the course It is a fun way for the students to solidify the geometric meaning of af ne maps and to introduce them to Maple Go to my directory korevaarfractals to see what this module is about This method of fractal generation has an interesting theoretical backing based on the contraction mapping principle from analysis and on Hausdorff distance between compact sets I would be happy to provide you with material for this module or even to present it in your class if you ask Next Bretscher very quickly introduces subspaces span independence basis and dimension in the context of Euclidean space before moving to abstract vector spaces This material will be used extensively in the 2280 semester Orthogonality in Euclidean space is discussed including Gram Schmidt orthogonalization pro jection and the the method s of least squares for nding optimal approximate solutions to overde termined systems as well as to data tting The idea of of Euclidean orthogonality generalizes to inner product spaces such as the L2 function spaces I mention Fourier series in this context so that they seem a little less mysterious in Math 2280 Bretscher holds off until chapter 6 to explain determinants and includes a good section on their geometric meaning I would mention the multivariable change of variables formula in integral calculus in this context it is not treated very completely in our multivariable calculus course Chapter 7 is devoted to eigenvalues and eigenvectors The material is motivated with interesting discrete dynamical systems This is a good time to review complex numbers and complex linear algebra as does the text If you have time and an energetic class it could also be appropriate to talk about Jordan canonical form for non diagonalizable matrices although I wouldn t try to include proofs of everything In chapter 8 the text uses eigenvector analysis to prove the spectral theorem about diagonalizing symmetric matrices and applies this result to quadratic forms and hence to conics and quartic surfaces 1 use this opportunity to talk about the multivariable second derivative test as well The nal section explains the singular value decomposition of a linear transformation between Euclidean spaces This decomposition has many applications in numerical analysis 8 2270 suggested lectures The following estimates add up to 48 lectures In the fall resp spring of the typical academic year there are 58 resp 57 MTWF class meetings leaving time for supplementary topics Maple labs exams and reviews in theory Feel free to modify these recommendations based on your own inclinations but don t shortchange the core material You will nd that when Bretscher comes to a complicated theoretical concept coordinates with respect to a basis comes to mind his presentation is very concise This means you will have to add material and examples and time for your class to really understand it 0 Chapter 1 Linear Equations and Matrices 4 lectures 0 Chapter 2 Linear Transformations 5 lectures Chapter 3 Subspaces of Euclidean Space and dimension 6 lectures Chapter 4 Linear ie vector Spaces 5 lectures 0 Chapter 5 Orthogonality and Least Squares 8 lectures Chapter 6 Determinants 5 lectures Chapter 7 Eigenvalues and eigenvectors 9 lectures Chapter 8 Symmetric matrices and quadratic forms 6 lectures 9 2270 computer projects These projects will be assigned to enhance the course material Aim for two or three substantive projects but you need not use mine In addition to the projects I expect to work Maple examples into class lectures and regular homework Project topics I have used in the past 0 Fractal generation by iterated function systems of af ne maps relates to the geometric interpretation of linear transformations chapter 2 This project also included an introduction to Maple and the Math lab 0 Least square data tting and function orthogonality topics from chapter 5 I like having the students collect height weight data and using it to derive an empirical power law from the ln ln data They always get a power law for weight as depending on power of height with power between 235 and 27 and national data indicates that 26 would be good power 0 Conic sections and quadric surfaces chapter 8 Mladen Bestvina did an interesting project in spring 2002 with discrete dynamical systems chapter 7 He s delinked the project but if you contact him I m sure he can point you where to go Let me know if you come up with successful projects so that I can pass your ideas on to future instructors 10 Math 2280 details The semester begins with rst order differential equations their origins geometric meaning slope elds analytic and numerical solutions The logistic equation and various velocity and acceleration models are studied closely The next topic is linear DE s of higher order with the principal application being mechanical vibrations friction forced oscillations resonance At this point we show how various scienti c models of dynamical systems lead to rst order systems of differential equations and then we complete the theoretical study of linear systems of DES which was begun in 2270 The concepts of the phase plane stability periodic orbits and dynamical system chaos are introduced with various ecological and mechanical models The study of ordinary differential equations concludes with an introduction to the Laplace transform The nal portion of Math 2280 is an introduction to the classical partial differential equations the heat wave and Laplace equations and to the use of Fourier series and separation of variable ideas to solve these equations in special cases Time permitting one can also introduce the Fourier transform 11 2280 suggested lectures The following estimates add up to 50 lectures In the typical fall spring term there are 58 resp 57 MTWF class meetings leaving some time for Maple labs reviews and exams If you nd this schedule too tight you might omit 37 and 97 for example Do not short change chapters 1 6 and try to spend enough time on chapter 9 so that students get some feel for partial differential equations 0 Chapter 1 Firstorder differential equations 5 lectures 7 11 Differential equations and mathematical models 7 12 lntegrals as general and particular solutions 7 13 slope elds and solution curves 7 14 Separable equations and applications 7 15 Linear rst order equations 0 Chapter 2 Mathematical Models and Numerical Methods 5 lectures 7 21 Population models 7 22 Equilibrium solutions and stability 7 23 Acceleration Velocity models 2 lectures 7 24 26 Numerical methods survey 0 Chapter 3 Linear Equations of Higher Order 8 lectures 7 31 32 lntroduction second order linear equations and general solutions of linear equa tions 7 33 Homogeneous equations with constant coef cients 2 lectures 7 34 Mechanical vibrations 7 35 Nonhomogeneous equations and the method of undetermined coef cients 7 36 Forced oscillations and resonance 2 lectures 7 37 Electrical circuits 0 Chapter 4 Introduction to Systems of Differential Equations 2 lectures 7 41 First order systems and applications 7 43 Numerical methods for systems 0 Chapter 5 Linear Systems of Differential Equations 7 lectures 7 51 52 General theory and the eigenvalueeigenvector method for homogeneous systems 2 lectures 7 53 Second order systems and mechanical vibrations 7 54 Multiple eigenvalue solutions 7 55 Matrix exponentials and linear systems 2 lectures 7 56 Nonhomogeneous linear systems 0 Chapter 6 Nonlinear Systems and Phenomena 6 lectures 7 61 62 Stability and the phase plane7 and linearization 2 lectures 7 63 Ecological Applications Predators and Competitors 2 lectures 7 64 Nonlinear mechanical systems 7 65 Chaos in dynamical systems 0 Chapter 7 Laplace Transform 5 lectures 7 71 Laplace transforms and inverse transforms 7 72 Transformation of initial value problems 7 73 Translation and partial fractions 7 74 Derivatives7 integrals7 and products of transforms 7 76 lmpulses and delta functions 0 Chapter 9 Fourier Series Methods 12 lectures 7 91 Periodic functions and trigonometric series 2 lectures 7 92 General Fourier series and convergence 7 93 Even odd functions and termwise differentiation 2 lectures 7 94 Applications of Fourier series to forced oscillation problems 7 95 Heat conduction and separation of variables 2 lectures 7 96 Vibrating strings and the onedimensional wave equation 2 lectures 7 97 Steady state temperature and Laplace s equation 2 lectures 12 2280 computer projects These projects will be assigned to enhance the course material Aim for at least 3 substantive projects in addition to regular homework which has computational aspects Possible topics for 2280 projects include Slope elds Euler s method for 1st order differential equations Chapters 1 2 Newton s law of cooling for a house Chapter 1 modeling populations with the logistic equations Chapter 2 modeling springs damping forced oscillations resonance and approximate resonance Chap ter 3 Earthquakes and multi story building Vibrations Chapter 5 pplane investigation of non linear phase portraits Chapter 6 Numerical methods for systems chaos in Duf ng s spring equation Chapter 6 Fourier series Chapter 9 Fourier series solutions to the heat and wave equations Chapter 9 NATIONAL HEIGHTWEIGHT DATA AND39HHEBNHINDEX There is a database at the US Center for Disease Control of national body data collected between 1976 and 1980 From this data I have extracted the median heights and weights for boys and girls age 219 Here is the national data heights are given in inches and weights are in pounds gt withlinalgwithplots l l gt BGaugmentagesAlA2 gt AstackmatrixtitleBG gt titlematrixl5 age boy height weight girl height weight gt Altransposematrix3593894l944347249651453655 7 573598628660673684689696696 29834l388428486548608665768823938 106812431326l42ll45l15531532 boy heights weights medians for ages 2 19 gt A2transposematrix3543844ll43946648951453l55 7 58261062663364264364264l645 2803263684l847052560865576l890lOOl 1081117lll761226128812451260 girl data gt agesmatrix18l2345678910111213l415l6 171819 boys girls combined gtage boy height weight girl height weight 2 359 298 354 280 3 389 341 384 326 4 419 388 411 368 5 443 428 439 418 6 472 486 466 470 7 496 548 489 525 8 51 4 608 51 4 60 8 9 53 6 665 53 1 65 5 A 10 55 7 768 55 7 761 11 57 3 823 58 2 89 0 12 598 938 610 1001 13 628 1068 626 1081 14 660 1243 633 1171 15 673 1326 642 1176 16 684 1421 643 1226 17 689 1451 642 1288 18 696 1553 641 1245 7 gt19 696 1532 645 1260 E When we do the lnln analysis of this data we nd that the least squares line t is gt lnw 2 593488078lnx 6 037404653 Here is a picture of the least squares line and the lnln data national nhtInwt data You notice that until adolescence the boy and girl data are more or less indistinguishable The quotbaby fa quot of small children may explain why the 2year olds are slightly above the line and the peaking near adulthood for both the males and females is quite likely to be the effect of their respective hormones But this data IS very close to a linear fit One might wonder what sort of biological advantage this sort of scaling is a consequence of I do Going back to the power law gt p 2 593488078 Czexp 6 037404653 fx gtCx p And here s a picture of the experimental power law graphed with the actual heights and weights National Power Law 160 140 120 100 80 60 4o Correlation coef cient In statistics one measures for a possible linear relationship between variables by using the sample correlation coefficient In class and in the text page 189 we discussed the fact that the correlation coef cient is really the cosine of the angle between the normalized vector of Xvalues and the normalized vector of yvalues The normalization was to subtract off the average values of each data set so that the averages were both zero A correlation coef cient near 1 implies high positive correlation If you compute the correlation coef cient for our lnln data dotprodx y costheta x y gt normx 2 normy 2 gt costhetaxvect xvectavyvect yvectav this is the sample correlation coefficient for our ln ln data 9924868726 gt evalf180Pi arccos the angle between our normalized deviation vectors in degrees 7027814524
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