HW#2, due 1/26. Selected solutions.

HW#3, due 2/2. Selected solutions.

HW#4, due 2/9. Selected solutions.

HW#5, due 2/16. Selected solutions.

HW#6, due 2/23.

HW#7, due 3/2.

HW#8, due 3/9.

HW#9, due 3/16 at 9:00 AM.

Office hours: Mon, Wed, 1:00 - 2:30, Room 383M.

Office hours: Tuesday, 3-5; Wednesday, 4-6; Thursday, 3-5, room 380G.

A secondary goal is to work on mathematical reasoning and proofs. Proofs are an essential technique in higher mathematics, and I feel that even for someone who does not want to become a mathematician, a facility with proofs is one of the most important things one can learn from the mathematics department. In elementary math courses one has to find answers; in advanced courses one also has to find reasons and explanations. This is a fun game, full of beautiful tricks, but it can be difficult for beginners. In this course we will do some explicit work on proof techniques, to try to provide a transition to more advanced mathematical courses.

This book has the advantage of being comprehensive and sophisticated, with lots of detailed explanations and examples and good problems. We will cover approximately 50% of its contents, and it should be of continued use to you in your future studies. A disadvantage is that some of the basic concepts might be a little hard to pick out amidst all the other stuff that is going on. This is one reason why it is important to come to class, where I will try to explain what I consider to be the most fundamental ideas, in the simplest possible manner. The book should be regarded as a resource for more comprehensive explanations, examples, and further directions.

- (1/10 - 1/12) Solving systems of linear equations. (Section 1.4
and Chapter 3)
- Elimination, reduced row-echelon form.
- Basic theorems on the existence and uniqueness of solutions.
- Matrix notation.

- (1/17 - 1/31) The geometry of vectors.
- Linear combinations and span. (Section 1.4)
- Subspaces of
**R**^n. (Section 1.3) - Linear independence, basis, and dimension. (Sections 1.5 and 1.6)
- The null space and column space of a matrix. (Section 2.1)
- Affine subspaces and the set of solutions to Ax=b. (Section 3.3)
- Inner products, orthogonal projection, Cauchy-Schwarz and triangle inequalities. (Section 6.1)
- Application: least-squares approximation. (Section 6.3)
- Orthonormal bases and Gram-Schmidt orthogonalization. (Section 6.2)
- Vector spaces in general. (Section 1.2)

- (2/2 - 2/14) The algebra and geometry of linear transformations.
- Representing linear transformations from
**R**^n to**R**^m by m by n matrices. (Section 2.2) - Examples: rotations, reflections, projections.
- Composition of linear transformations and matrix multiplication. (Section 2.3)
- Injections, surjections, isomorphisms. (Appendix B, Section 2.4)
- Inverses. (Section 3.2)
- Representing linear transformations in different coordinate systems. (Sections 2.2 and 2.5)

- Representing linear transformations from
- (2/16 - 2/23) Determinants. (Chapter 4)
- Axioms for determinants.
- Further properties and computation.
- Cramer's rule.
- Geometric meaning of determinants: volume distortion, preservation of orientation, invertibility.
- Existence of determinants.

- (2/26 - 3/12) Deeper understanding of linear transformations.
- Eigenvectors and eigenvalues. (Section 5.1)
- Diagonalizability. (Section 5.2)
- Symmetric matrices are diagonalizable.
- Applications: systems of linear ODE's (Section 2.7), Markov chains (Section 5.3), the inertia tensor (if time permits - not in the book).

- (3/14 - 3/16) Summary and review.

Up to Stanford Undergraduate Mathematics.

updated: 2/23/01