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UMBC MATH 430 - Syllabus MATH 430

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Math 430, Fall 2008Matrix AnalysisInstructor: John ZweckOffice: MP 424Email: [email protected]: I will maintain a web page for the course, linked from my web pagewww.math.umbc.edu/∼zweck. I will also communicate with you us-ing a class email list.Phone: (410) 455 2424Fax: (410) 455 1066Lectures: MW, 2:30-3:45pm (BIOL LH1)Text: “Matrix Analysis and Applied Linear Algebra”, by Carl MeyerOther Books: You may also find the following excellent books useful, though I willnot explicitly rely on them.1. “Applied Linear Algebra”, by Peter J. Olver and ChehrzadShakiban Prentice-Hall, Upper Saddle River, N.J., 20052. “Linear Algebra and Its Applications”, by Gilbert Strang3. Appendix on Matrix Theory in “An Introduction to Multi-variate Statistical Analysis”, 3rd edition, by T.W. Anderson4. “Linear Algebra and its Applications”, by David C. Lay(This is the Math 221 book)5. “Matrix Computations”, by G. Golub and C.F. Van LoanPrerequisites: Math 221, Math 251 and Math 301 or permission of instructor.Note that you will be expected to do some proofs and wewill discuss proofs a lot in class.Your Review: I will assume that by Wed Sept 3rd you have reviewed the followingsections from Meyer’s book: 1.2, 1.3, 2.1-2.5, 3.2, 3.5. The materialis basically the same as that covered in Sections 1.1-1.8, 2.1-2.3 ofLay’s book (the Math 221 text). We will use this review materialextensively in our course, but I will not discuss it in class. I willassign homework on this material.Material Covered: In class we will cover the following sections of Meyer’s book:Chapters 3.2-3.7, 4.1-4.8, 5.3-5.6, 5.8-5.11, 6.1, 6.2, 7.1-7.6.Office Hours: M 10:00-11:00, W 11:00-12:00 and by appointment. I can often an-swer short questions immediately after class. If you cannot cometo my office hours please contact me in class or by email to set upa time to meet. Also, you are encouraged to ask me questions byemail. I rarely check my phone messages.1Course Summary and Learning GoalsIn this course we study some of the basic techniques from linear algebra that undergird theanalysis and simulation of a large and diverse collection of applications in science and engi-neering. The equations used to model these applications typically involve many variables.Even when the underlying equations are nonlinear, solution techniques—both analytical andnumerical—often involve exploiting the geometric and algebraic structure of linear transfor-mations on finite dimensional vector spaces. Since every such linear transformation is givenby matrix multiplication, this involves analyzing the structure of matrices.The main themes of the course are: (1) Matrix algebra (akin to vector algebra in Multi-variable calculus); (2) Linear transformations, focusing on the rank-and-nullity and change-of-basis theorems; (3) Linear transformations that preserve the inner product between vec-tors, with an emphasis on the discrete Fourier transform; (4) A careful development of thetheory and meanings of determinants; (5) Eigensystems with a focus on the spectral theoremfor normal matrices, functions of matrices and with applications to systems of ordinary dif-ferential equations; (6) Some general decomposition theorems for matrices; (7) Calculationsinvolving block matrices.The course is designed to serve the needs of undergraduate Mathematics, ComputerScience, and Physics majors, as well as incoming graduate students in Statistics and Math-ematics. The course material is widely used in pure and applied mathematics and statistics,and in applications fields. Applications include solutions of systems of linear ordinary differ-ential equations, methods to solve large linear systems—and therefore linear and nonlinearpartial differential equations—on a computer (see Math 630), multivariate probability andstatistics, signal and image processing, quantum mechanics, Hilbert space theory (lineartransformations on infinite dimensional inner product spaces), and differential geometry (seeMath 423).Homework and exams will emphasize calculations for specific examples based on thetheory discussed in class, as well as some more abstract proofs based on similar sorts ofcalculations to the ones performed in class. Exams will also test your understanding ofdefinitions and theorems covered in class.Students wishing to master the course material will be guided by the following learninggoals.1. Students will master the definitions, examples, calculations, theorems and proofs dis-cussed in class and covered on homework.2. Students will organize their understanding of the course material using the seventhemes discussed above, and identify interconnections between them.3. Students will become proficient at setting up and performing matrix algebra calcula-tions to study the structure of matrices, and apply these technqiues throughout thecourse.4. Students will understand and be able to use the algebraic and geometric structuresof special classes of matrices and linear transformations, including matrices that aresquare, diagonal, symmetric, block, orthogonal, unitary, Fourier, diagonalizable, nor-mal, positive definite, or stochastic.25. Students will do a class project in which they will gain experience in (1) reading andassimilating the material in a research-level journal article; (2) gaining a higher-levelunderstanding of a sophisticated real-world application of the course material, ratherthan simply tackling isolated problems; (3) formal mathematical and technical writ-ing; (4) working indepenedently of the course instructor in collaboration with anotherstudent.6. Students will use the experience gained in this course to prepare them for higher levelmathematics courses, for research, and for professional work involving mathematics.Specifically, they will learn how how mathematics involves gaining mathematical andscientific knowledge by integrating concepts from a wide variety of subjects. For exam-ple, they will see how matrix analysis integrates ideas from linear algebra, calculus, thegeometry of vectors and vector subspaces, differential equations, analysis and physics.In particular, students will learn to identify the abstract mathematical principles thatare encoded into matrix analysis and which unite apparently disparate examples andapplications.7. Finally, students will be encouraged to develop an appreciation for the many applica-tions of matrix analysis.Academic MisconductI will not tolerate cheating in any form. All instances of


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