Textbook: “Linear Algebra” by Friedberg, Insel and Spence (4th edition).

I will use my own lecture notes which follow the course outline. It is not necessary, but recommended, that you read the textbook concurrently and come to office hours if there is anything you do not understand.

Exams:

• Midterm: Friday 05/05/2006, 11:00am – 12:00pm.
• Final: Thursday 06/15/2006, 11:30am – 2:30pm.

Lecture Notes, Homeworks, and Exams

1. Homework 1 (Due 04/13), Introduction and Overview, Linear Vector Spaces.
2. Homework 2 (Due 04/20), Setting up a coordinate system.
3. Homework 3 (Due 04/27), Linear Transformations.
4. Homework 4 (Due 05/04), Midterm Review Problems, Matrix Representations of Linear Transformations, Midterm Review.
5. Invertibility and Isomorphisms, Midterm (Solution).
6. Homework 5 (Due 05/18), Change of Coordinates, Determinants.
7. Homework 6 (Due 05/25), Eigenvalues and Eigenvectors.
8. Homework 7 (Due 06/01), Diagonalizability.
9. Homework 8 (Due 06/08), Inner Product Spaces.
10. Final Review Problems, Linear Operators, Final Review.

• Homeworks and lecture notes will be posted at the beginning of each week.

  The homeworks should be turned in to the TA in the Thursday discussion section on the following week (So you have about 10 days to think about a homework).

  No late homework will be accepted, unless with very good reasons. Please talk to the instructor before the homework is due if you have to miss the due date of that homework.

  The lowest homework score will be dropped.

• The exams will be open notes. Books, calculators or laptops are not allowed.
• Please remember to bring a picture ID to the exams.

Grading:

• my.ucla.edu will be used.

• Two schemes will be used.

  1. Homework: 20%; Midterm: 30%; Final: 50%.
  2. Homework: 20%; Final 80%.

  The final grade will be the higher one from the two methods.

• As a consequence, no make-up for midterm under any circumstances.
• The final will cover all the materials. With about 35% points on materials before the midterm and 65% on those after.

• The grade “incomplete” will only be given in very special cases, when a student

  • is doing well in the course, and
  • cannot take the final for a very serious and documented reason.
  • contacts the instructor before the final.

Further readings:

• How did linear algebra evolve over the last 4,000 years.

  1. Chiu Chang Suan Shu (The Nine Chapters on the Mathematical Art),

     Chapter 8 “Rectangular arrays” (System of linear equations). (In Ancient Chinese). (The full book is available here. English description of the book.)

     • An English monograph: The Nine Chapters on the Mathematical Art, by K. Shen, J. N. Crossley and A. W.-C. Lun, Oxford University Press and Science Press, 1999. Sample pages about the first problem of Chapter 8 (the one in my first lecture).

  2. History of Algebra.

     From The MacTutor history of mathematics archive. Nice description of the evolution of linear spaces and matrices/determinants.

  3. Kline, Morris. Mathematical Thought from Ancient to Modern Times,

     Sections 1.5 (Babylonian algebra), 2.3 (Egyptian algebra), 25.3 (determinants) and Chapter 33 (19th century).

  4. Dieudonné, Jean Alexandre. History of Functional Analysis.

     Chapter IV (from finite to infinite dimensional “linear algebra”).

• Analytical properties of linear transformations (Operator theory).

  We have seen that we can assign norms to vectors, which allows us to estimate, approximate, etc.. It turns out that we can do the same thing to linear transformations. And furthermore, we can apply theories from calculus and complex analysis to study linear transformations...

  1. Kato, Tosio. Perturbation Theory for Linear Operators, Chapters one and two. These two chapters are about linear transformations in finite dimensional vector spaces.

• Numerical linear algebra.

  It's one thing to know that a linear system has a unique solution, it's a very very different thing to actually find it...

  1. Trefethen, L. N. and Bau, David. Numerical Linear Algebra. Basics of numerical linear algebra in 40 short and easy to understand lectures. Highly recommend.
  2. Higham, N. J. Accuracy and Stability of Numerical Algorithms. What can go wrong in numerical computations and why.