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Syllabus AUBG, Math & Sci Dept, Spring 2012 MAT 105: Linear Algebra required for Math Major & Minor; GenEd in quantitative reasoning prerequisites: appropriate score on SAT Math test or MAT 100 Introductory Mathematics Alexander GANCHEV e-mail: [email protected], web-space: http://home.aubg.bg/faculty/aganchev/ , office 304 (BAC), phone: 480, office hours: TW 13:00-14:15 and by appointment classes meet: TF 14:15-15:30, room 201 BAC Course description: The course is an introduction to the theory of linear (i.e., vector) spaces and linear transformations between them (i.e., matrices). Course objectives (Student Learning Outcomes): In this course one has to learn the GaussJordan elimination method. We will be using it from the first day to the last. The course is difficult because there are numerous abstractions we have to become well familiar with – the algebra of matrices (a non-commutative algebra), translation of geometric data into algebraic, vector spaces and linear transformations between them, basis and dimension of a vector space, eigenvectors and eigenvalues of matrices and diagonalization of matrices. The course covers the learning outcomes for “quantative reasoning” (see the Catalog, section Degree Requirements) the most important one of these being “critical thinking”. Textbook: David C.Lay, Linear Algebra and its applications , 3rd edition (Pearson, 2006) Supplementaty texts (on reserve in the library): G. Strang, Linear Algebra and its Applications; H. Anton and C. Rorres, Elementary Linear Algebra, applications version Assessment: Your grade will be formed by short quizzes homework/attendance/oral exams midterm exam final exam total 60 points = 20 % 30 10 % 90 30 % 120 40 % ____________ 300 points = 100 % The duration of a short quiz will be from 10 to 30 min and will consist of routine problems. The duration of a midterm exams will be a full class period and may consist 6 or more problems (some will be routine but do expect also several nontrivial problems). The final will be comprehensive, i.e., over all the material covered by the course. For the midterm exam there will be one, and only one, makeup. Percents/Grade Map: D≥45, D+≥55, C-≥60, C≥65, C+≥70, B-≥75, B≥80, B+≥85, A-≥90, A≥95 % Exam policies: During quizzes, exams and the final all that you will need and will be allowed to use is a pen/pencil and a notebook that I will give you (no textbooks, notes, calculators, mobile telephones or other electronic gadgets, sheets of paper etc., no sheets of paper flying around the room, etc.). I will assign seats before the exam, i.e., before the exam you may help me pull the tables apart and wait to be assigned a seat. You should work strictly by yourself – you should not communicate in any way with your classmates – violation of this will be considered cheating with all the ensuing consequences (see the AUBG documentation for the consequences of cheating). Cheating is not only talking to the person next to you (talking about anything: math, the problems, the weather, last nights party …) but also intentionally making your work available to others during the exam. One or two weeks after the midterm there will be a makeup for that exam. It will be scheduled on Tuesday evening (7pm or later) or on Friday from 5:30pm. This is your second chance. Under no circumstances will there be a third chance! Attendance: Students are expected to attend classes regularly and should comply with the university attendance policies. I expect you to come to class prepared (having read the assigned text if there is such) and to show active participation during the lecture. A student missing, without good reason, more than three classes may be dropped from the class. Assignments: Often I will assign sections from the textbook for you to read ahead and from time to time I will make reading quizzes to check if you have read the assigned part. I will also expect that on the average you spend more than 4 hours per week (on top of the regular classes) working on problems from the book (this is on the average – the need for this extra work is very individual). I will give a list of problems for optional homework, i.e., I will not collect most of these optional homeworks (from time to time I will collect samples) but you are strongly encouraged to do as many as you can. Office hours: If the “official” office hours are not convenient for you please contact me to arrange some other time. Don’t be afraid to come and ask. There are no stupid questions. General advise: Don’t be surprised to find out that a math course in College is very different from your High School experience. At school one topic (say the quadratic equation) is studied for many weeks or even months – here we go much faster – if I say 5 or 10 times faster probably I won’t be exaggerating. The only way to manage with this pace is to work individually outside of class. If you are not a mathematical genius but want to be successful in the course probably you have to work at least 4 hours per week besides attending classes and actively participating in them. While on exams you work strictly by yourself I would encourage you to get together in small groups to work outside of class. The textbook and notes taken in class are not substitutes for one another – rather they should complement one another. If you understand well the concepts and practice enough the techniques then you will feel comfortable with any exam. Disclaimer: This syllabus is subject to modification. The instructor will communicate with students on any changes. The distribution of weeks per chapter is only approximate. tentative schedule (this is an indication of the topics covered but not the order – the chapter on eigenvalues is so important that we will take it as early as possible): I. Systems of linear equations, Matrices and Gaussian elimination: 4 lectures [1] Chapter 1, 1.1, 1.2, 1.3, 1.4, 1.5, 1.7, 1.8*, 1.9* Systems of linear equations [1, 1.1 ] Gaussian elimination -Row reductions and echelon forms [1, 1.2] Vectors in R^n, linear combinations, Span {v_1, …, v_k} [1, 1.3] The matrix equation Ax=b [1, 1.4] Solution sets of linear systems [1, 1.5] Linear independence [1, 1.7]* Linear Transformations. The matrix of a linear transformation [1, 1.8, 1.9] * II. Matrix Algebra 4 lectures [1] Chapter 2, 2.1, 2.2, 2.3, 2.8, 2.9, Matrix operations [2, 2.1] The inverse of a matrix. Elementary matrices [2, 2.2] Characterization of invertible matrices [2, 2.3] Subspaces of R^n. Column space and nullspace of a matrix. [2, 2.8] Dimension and rank, the rank theorem, the basis theorem. Rank and the invertible matrix theorem.* [2, 2.9] III. Vector spaces 5 lectures [1], Chapter 4, 4.1, 4.2, 4.3, 4.5 , 4.6, 4.4*, 4.7* Vector spaces and subspaces [4, 4.1] The Four Fundamental Subspaces of a matrix [4, 4.2] Linearly independent sets, bases, and dimension [4, 4.3] The dimension of a vector space [4, 4.5] Rank (The Fundamental Theorem of Linear Algebra, Part I) [4, 4.6] IV. Determinants: 3 lectures [1], Chapter 3.1, 3.2, 3.3, 2.3, 2.8, 2.9 Introduction to determinants, Properties of the Determinant Applications of determinants : Cramer’s rule, volume. V. Eigenvalues and eigenvectors 4-5 lectures [1], Chapter 5, 5.1, 5.2, 5.3, 5.4, Chapter 7, 7.1 Eigenvectors and eigenvalues [5, 5.1] The Characteristic equation [5, 5.2] Diagonalization, diagonalizing matrices [5, 5.3] Diagonalization of symmetric matrices * [7, 7.1] Eigenvectors and linear transformation * *[5, 5.4] VI Orthogonality: (time permiting) [1], Chapter 6, 6.1, 6.2, 6.3, 6.4 2 lectures , Inner product, length and orthogonality. Orthogonal sets Lines and planes in 3 –space Orthogonal bases* (The following schedule is only tentative – there will be some, hopefully not big, deviations.) # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 day date T 17.01 F 20.01 T 24.01 F 27.01 T 31.01 F 03.02 T 07.02 F 10.02 T 14.02 F 17.02 T 21.02 F 24.02 T 28.02 F 02.03 T 13.03 F 16.03 T 20.03 F 23.03 T 27.03 F 30.03 T 03.04 F 06.04 T 10.04 F 13.04 T 17.04 F 20.12 T 24.12 F 27.12 section 1.1,2 1.3 1.5 1.7 1.8 1.9 2.1 2.2,3 2.8 2.9 3.1,2 3.3 5.1 Quiz 1 (on chap 1) Quiz 2 (on chap 2) withdrawal deadline BREAK Review Mid Term 5.2 5.3 6.1 6.2 6.3 6.4 6.5,7 4.1,2,3 4.4 4.5,6 Quiz 3 Good Friday Holiday Quiz 4 Review FINAL EXAMS