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MATHEMATICS 2270-1Linear AlgebraFall semester 2005text:when:where:Linear Algebra with Applications, edition 3Eby Otto BretscherMTuWF 9:40-10:30JTB 120instructor:office:telephone:email:office hours:problem session:Prof. Nick KorevaarLCB [email protected] .e duM 1-1:50 p.m., T 11-12 a.m., W 1-1:50 p.m.Th 9:40-10:30, in LCB 121course web page: www.math.utah.edu/∼koreva ar/2270fall05prerequisites: Math 1210-1220, or Math 1250-1260, or Math 1270-1280; first year Calculus.Previous exposure to vectors, eithe r in a multivariable Calculus course (e.g. 2210 or 1260 or 128 0)or in a Physics course, is helpful but not essential.course outline: This is the first semester in a year-long sequence devoted to linear mathematic s.Our topic this semester is linear algebra, a fundamental area of mathema tics that is used to describeand study a multitude of subjects in science and life. The origins of th is field go back to the algebrawhich one must solve to find the intersection of two lines in a plan e, or of several planes in space,or more genera lly the solution set of one or more simultaneous “linear” equations involving severalvariables.We shall cover chapters 1-8 of the text. The detailed syllabus which follows the course summa rybelow is an educated guess at how we will procee d, although the only things for certain are theexam dates.The course begins in chapter 1 by studying linear systems of equations and the Gau ss-Jordanmethod for systema tically solving them. Linear algebra always has a “linear geometric” interpre-tation, which may have been omitted in your earlier exposure. We study aspe cts of this lineargeometr y for the Euclidean plane in chapter 2, as well as the relation betwee n inverse matricesand inverse transformations In chapter 3 we undertake a more systematic exploration of the linea rgeometr y related to transfor mations and sub spaces of Rn.The relat ively concrete concepts for subsp aces of Rnwhich we discuss in chapter 3, conceptsincluding spa n, independence, basis, dimension and coordinates, actually apply to many otherspaces, called “vector spaces” or “linear spaces”. These generalized notions have many commonapplications to seemingly diverse are as of mathematics, including the study of differential equationsin Math 2280. S o, in chapter 4 we study these notions abstractly.You know what it means fo r two directions to be perpendic ular, and may already have used the“dot product” to test for this condition. This notion of “orthogonality” is a major theme of linearalgebra, and is the focus of chapter 5. We will study orthogonal projections and transformations,Gram-Schmidt orthogon alization, methods of least squares, notions of orthogon ality for function s,Fourier series.You have probably used determinants as a computational tool in high school algebra, butare p robably not aware of all their uses and why their magic properties work. We will stud ydeterminants in deta il in chapter 6, including t heir important geometric meaning related to orientedareas and volumes.1There are special vecto rs known as eigenvectors, related to the geometry of linear transforma-tions . They also ar ise in the study of dynamical systems and in differential equations. Eigenvectorsand eigenvalues are the topics of chapter 7. In chapter 8 we will see some initial applications ofeigenvecto rs, r elated to c onic sections, quadric surfaces, and the multivariable sec ond deriva tivetest. Finally, in section 8.3 we will discuss the singu lar va lue decomposition for arbitrary lineartransformations, impo rtant conc eptually and in numerical applications. In Math 2280 you will seemany more applications of eigenvalues and eigenvectors.computer projects: There will be approximately 3 computer projects during the semester, toenhance and expand upon the material in the text. They will be written in the software packageMAPLE. On MAPLE days we will meet in the Math Department Computer Lab in LCB 115.We do not assume you h ave had any previous exper ie nce with this software and we will make thenecessary introductions during the first lab project. You will also fin d MAPLE useful for some ofyour regular homework. There are many labs around campus where MAPLE is a lso available, forexample at the College of Engineering and Marriott Library, a s well as in the walk-in Math lablocated down the hall from LCB 115, in the Rushing Stu dent Center.tutoring center: The Math Department Tutoring Center offers free tutoring for all Math coursesthrough the 2000-level. It is located in the Rushing S tudent Center, in the basement between JWBand LCB. The tutoring center and th e adjoining walk-in computer lab are open from 8 a.m. to 8p.m. on M-Th, and from 8 a.m to 6 p.m. on Fridays, starting Wednesd ay August 31. Some, butnot all of the math tu tors welcome questions from Math 22 70 students; once you find a tu tor youlike you should learn their available h ours.grading: There will be two midterms, a comprehensive final examination, and homework. (Home-work assignments and other course information will be posted on the course web page.) Eachmidterm will count for 20% of your grade, homework (including book and Maple assignments) willcount for 30%, and the final exam will make up the remaining 30%. Homework assigned by Fridayof one week will be collected the following Friday, in o rder that it may be partially graded. Notethat in addition to the regular office hours and th e tutoring center, you may attend t he weeklyhomework problem session which I will lead, Thursdays in LCB 121 from 9:40-10:30. Map le projectswill generally be due one week after they a re assigned. A home work grader will partially gradeyour assignments. The value of carefully working homework problems is that mathemat ic s (likeanything) must be practiced and experienced to be learned.It is the Math Departme nt policy, and mine as well, to grant any withdrawl request until theUniversity deadline of Friday October 21.ADA Statement: The American with Disabilities Act requires that reasonable accomodationsbe provided for students with physical, sensory, cognitive, systemic, learning, and psychiatricdisabilities. Please contact me at the beginning of the semester to discuss any such accommodationsfor the co urse.2Tentative Daily Scheduleexam dates fixed, daily subject matter approximatedWFMTWFMTWFMTWFMTWFMTWFMTWFMTWFMTWF24 Aug26 Aug29 Aug30 Aug31 Aug2 Sept5 Sept6 Sept7 Sept9 Sept12 Sept13 Sept14 Sept16 Sept19


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U of U MATH 2270 - Syllabus

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