# Math 115AH: Honors Linear Algebra

Fall 2009

Time and Place: MWF 1-1:50 pm in MS 5118

 Instructor: Ciprian Manolescu E-mail: cm@math.ucla.edu Office: MS 6921 Office Hours: Mon 10-11am, Wed 2-3pm, and by appointment Section: TR 1-1:50pm in MS 5118 Teaching Assistant: Anush Tserunyan E-mail: anush@math.ucla.edu Office Hours: Tue 3-4pm and Thu 4-5pm in MS 2361

Web page: http://www.math.ucla.edu/~cm/115ah.html

Prerequisites: Math 33A (basic linear alegbra) or equivalent. You should be able to manipulate matrices and determinants, and solve systems of linear equations. Familiarity with complex numbers is also required.

Course description: This is an honors introductory course in abstract linear algebra. Although we use the same textbook as the sections of Math 115A, we will cover more material, and we will put more emphasis on doing rigorous proofs. The course should provide a valuable introduction to mathematical thought.

Topics to be covered are: vector spaces, subspaces, basis and dimension, linear transformations and matrices, rank and nullity, change of basis and similarity of matrices, dual spaces, direct sums, eigenvectors and eigenvalues, inner product spaces, orthogonality and orthonormality, Gram-Schmidt process, adjoints of linear transformations, quadratic forms and symmetric matrices, orthogonal and unitary matrices, spectral theorem.

Textbook: S. H. Friedberg, A. J. Insel and L. E. Spence, Linear Algebra, Custom UCLA Edition, Pearson Prentice Hall, 2003. The plan is to cover Chapters 1, 2, 3, 4, 5.1-5.2, and 6.1-6.6 from the textbook. Note that the Custom UCLA Edition differs from the standard Fourth Edition (available at various online bookstores) in that it contains an appendix about how to write proofs.

Another recommended resource is the book by Peter Petersen, Linear Algebra, available online.

Grading: 50% final, 25% best score of two midterms, 25% homework.

Exams: There will be two in-class midterms: one on Monday, October 19, and the other on Wednesday, November 18. The final exam is scheduled for Friday, December 11, 8:00 AM - 11:00 AM.

No make-up exams will be given, except in case of a documented emergency.

Homework: Homework will be assigned every week and will be due the following Friday. There will be a total of eight problem sets. The lowest score of the eight will be dropped, and the other seven will be added up to give your total homework score.

The homework assignments will be handed out in class and will also be posted on the web page. You must hand in the homework in class each Friday. You are encouraged to talk about the problems with other students, but you should write up the solutions individually. You should acknowledge the assistance of any book, student or professor. No late homework will be accepted.

Material for Midterm 1: Sections 1.1-1.6, 2.1-2.2, and the Supplement to the textbook. Practice Midterm 1 (with solutions).

Material for Midterm 2: Sections 2.3-2.6, 5.1-5.2, 6.1-6.2. Some concepts from Chapters 3 and 4 may also be tested, particularly from Section 3.2. Practice Midterm 2 with solutions.

The Final Exam will cover all the material for the two midterms, plus Sections 6.3-6.6 from the textbook. The emphasis will be on the material after the second midterm. Practice Final. The actual final will consist of 20 True/False questions (one point each) and 8 problems (10 points each), for a total of 100 points.

Special office hours for the final week (C. Manolescu): Monday 10am-noon and 1-3pm, Thursday 3-4pm. The TA will hold a review session on Wednesday at 1pm.

The site with solution sets is password protected.