# Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

MAT 22A: Linear Algebra
Approved: 2012-09-01 (revised 2013-06-01, )

ATTENTION:

For information about the textbook, please consult with the assigned instructor for details.

Suggested Textbook: (actual textbook varies by instructor; check your instructor)
Introduction to Linear Algebra, 5th Edition by Gilbert Strang; Wellesley Cambridge Press; \$48.00-65.00.
Search by ISBN on Amazon: 978-0980232714

Suggested Schedule:

 Lecture(s) Sections Comments/Topics 1 1.1 Vectors and linear combinations. 2 1.2 Lengths and dot products. 3 1.3 Matrices. 4 2.1 Vectors and linear equations. 5 2.2 The idea of elimination. 6 2.3 Elimination using matrices. 7 2.4 Rules for matrix operations. 8 2.5 Inverse matrices. 9 2.6 Elimination = Factorization: A = LU 10 2.7 Transposes and permutations. 11 3.1 Spaces and vectors. 12 3.2 Nullspace of A: Solving Ax = 0 13 3.3 The Rank and the Row Reduced Form 14 3.4 The complete solution to Ax = b 15 3.5 Independence, basis, and dimension. 16 3.6 Dimensions of the Four Subspaces. 17 4.1 Orthogonality of the Four Subspaces. 18 4.2 Projections. 19 4.3 Least squares approximations. 20 4.4 Orthogonal bases and Gram-Schmidt. 21 5.1 The properties of determinants. 22 5.2 Permutations and cofactors. 23 6.1 Introduction to eigenvalues. 24 6.2 Diagonalizing a matrix. 25 6.4 Symmetric matrices. Time Permitting 6.5 Positive definite matrices.

Lectures 1 and 2 can be combined; Skip section 5.3 (Cramer’s Rule, Inverses, and Volumes); Skip section 6.3 (Applications of Differential Equations).

Learning Goals:

Linear Algebra is the foundation upon which the science and engineering of the modern digital age is built. Knowledge of linear algebra is a basic prerequisite for the study of Signal Processing, Statistics & Data Analysis, Ordinary & Partial Differential Equations, Quantum Mechanics, Quantitative Economics, Optimization & Control, Network, Numerical Analysis, Mathematical Finance, and Computational Models of Aerodynamics, Climate Change, Hydrology, Structures, Traffic Flow, Weather, etc., Indeed, it is difficult to find an area in engineering and science, including many of the social sciences, in which some essential aspect of that area does not require a background in Linear Algebra.

The purpose of MAT 22A is to introduce students to the fundamental objects and concepts in Linear Algebra, including scalars, vectors, matrices, diagonal matrices, symmetric matrices, inverse matrices, singular & nonsingular matrices, permutation matrices, linear combination, linear dependence & independence, vector spaces, subspaces, the dimension of a vector space, bases of vector spaces, rank, nullity, the four fundamental subspaces associated with a matrix, projections, determinants, permutations and cofactors, and eigenvectors & eigenvalues, and the fundamental operations on these objects including the dot product, matrix multiplication, matrix transpose, Gaussian elimination, Gauss-Jordon elimination, LU decomposition, matrix factorizations, reduction of a matrix to row reduced echelon form (RREF), the solution of systems of linear equations in an arbitrary number of unknowns, the least squares procedure and the Gram-Schmidt procedure. In MAT 22A a modern, innovative approach is taken in presenting this material. Each new idea is introduced by constructing concrete examples, in which, for example, the student learns to determine key parameters associated with a given matrix, (e.g., its rank and nullity), by applying procedures, (e.g., row reduction) to find the relevant canonical form of the matrix (e.g., RREF) from which these parameters can be easily found. Furthermore, this approach provides the student with a constructive proof of each of the key theorems presented in the course. This allows students to learn the theory underlying Linear Algebra by associating facts (i.e., theorems) with procedures they have learned to perform on matrices and vectors, such as reduction to RREF. Thus, this approach gives students an introduction to logic and proof, which is one of the goals of this course.

Students who master the material in 22A will have the background necessary to begin study of a wealth of topics in the sciences and engineering, including, but not limited to, Digital Signal Processing (discrete Fourier Series), Graphs and Networks (leading to the edge-node matrix for Kirchhoff’s Laws), Markov Matrices (as in Google’s PageRank algorithm), Linear Programming (an understanding of which is essential in Quantitative Economics and Financial Mathematics), Matrices in Statistics and Probability, and Computer Graphics. (This is a partial list of the topics covered in the eighth chapter of the textbook. The first five and one-half chapters are covered in 22A.) Students who successfully complete MAT 22A will also have a working knowledge of the computational tool MATLAB and how to use it to compute the solutions to the problems encounter in 22A. Students may obtain this knowledge by having successfully completed Engineering 6 prior to taking 22A or by taking MAT 22AL concurrently with 22A.

Assessment:

To assess the learning outcome for this course, there will be homework, including computational homework requiring students to use MATLAB, two midterms, and a final exam.