# Math 33A: General Course Outline

## Catalog Description

33A. Linear Algebra and Applications. (4)Lecture, three hours; discussion, one hour. Requisite: course 3B, 31B or 32A with a grade of C- 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. 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.

## Textbook

O. Bretscher, Linear Algebra, 5th Ed., Prentice Hall.

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, C. Jung, 9/14

## Schedule of Lectures

Lecture Section Topics

1-2

Chapter 1 (1.1-3)

Systems of linear equations, associated matrix equations, row reduction of a matrix, Gauss-Jordan elimination

3-6

Chapter 2 (2.1-4)

Linear transformations, invertible matrices, matrix algebra

7-10

Chapter 3 (3.1-4)

Subspaces of Rn, linear independence, row space, column space, bases, dimension, kernel and image of linear transformations, rank-nullity theorem, 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, applications to discrete dynamical systems,

24-26

Chapter 8 (8.1-3)

Diagonalization of symmetric matrices, applications to quadratic forms, SVD (singular-value decomposition)