# Math 33AH: General Course Outline

## Catalog Description

33A. Linear Algebra and Applications (Honors). (4) Lecture, three hours; discussion, one hour. Enforced requisite: course 3B or 31B or 32A with grade of B or better. Introduction to linear algebra: systems of linear equations, matrix algebra, linear independence, subspaces, bases and dimension, orthogonality, least-squares methods, determinants, eigenvalues and eigenvectors, matrix diagonalization, and symmetric matrices. Honors course parallel to course 33A. P/NP or letter grading.

## Course Information:

The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and two midterm exams. These are scheduled by the individual instructor.

The purpose of Math 33A is to provide mathematicians, engineers, physical scientists, and economists with an introduction to the basic ideas of linear algebra in n-dimensional Euclidean space. Abstract vector spaces are not covered; they are treated in Math 115A.

Students in the course should have covered the following topics in previous high school and college mathematics courses:

• solving linear systems of equations,

• matrices, matrix multiplication,

• two-by-two and three-by-three determinants,

• complex numbers,

• complex polynomials, the fundamental theorem of algebra.

This background material is reviewed in the course, though briefly.

The topics in linear algebra that are covered in Math 33A include:

• systems of linear equations, associated matrix equations,

• row reduction of a matrix,

• linear transformations,

• invertible matrices,

• subspaces, linear independence, bases, dimension,

• row space, column space, rank-nullity theorem,

• determinants,

• orthogonality, orthonormal bases,

• orthogonal matrices,

• Gram-Schmidt process, QR factorization,

• least-squares approximation, normal equations,

• eigenvalues, eigenvectors, similarity, diagonalization,

• applications to discrete dynamical systems,

• diagonalization of symmetric matrices,

• applications to quadratic forms, singular value decomposition.

## Textbook

O. Bretscher, Linear Algebra, 5th Ed., Prentice Hall. Check Schedule of classes for most current textbook.

Since the syllabus includes some important material for engineers at the end of the course (Chapter 8), the pacing of lectures is particularly important. Some time can be saved by synopsising the properties of determinants and leaving the details to the students. The students are already familiar with two-by-two and three-by-three determinants.

Most of the students are already familiar with matrix multiplication.

The ad hoc definition of "linear transformation" in Section 2.1 should be replaced by the correct definition, which can then be related to the definition given in the textbook.

Chapter 4 and Section 5.5 are generally not covered.

The QR decomposition in Section 5.2 is important for the engineers.

Most students will have seen the polar form of complex numbers given in Section 7.5 (in high school), but most students will not have seen the exponential form (Euler's formula) in previous courses.

Positive-definite matrices (Section 8.2) and the singular-value decomposition (Section 8.3) are very important for the engineers.

Outline update: T. Gamelin, 3/04

## Schedule of Lectures

Lecture Section Topics

1-2

Chapter 1 (1.1-3)

Linear systems, Gauss-Jordan elimination

3-6

Chapter 2 (2.1-4)

Linear transformations, inverses, matrix algebra

7-10

Chapter 3 (3.1-4)

Subspaces of Rn, linear independence, bases, dimension, kernel and image of linear transformations, coordinates

11-15

Chapter 5 (5.1-4)

Orthogonality, orthonormal bases, orthogonal projections, orthogonal transformations, orthogonal matrices, Gram-Schmidt process, QR-factorization, least squares methods

16-19

Chapter 6 (6.1-3)

Determinants

20-23

Chapter 7 (7.1-5)

Eigenvalues, eigenvectors, diagonalization of matrices

24-26

Chapter 8 (8.1-3)

Symmetric matrices, SVD (singular-value decomposition)