Math 415. Appled Linear Algebra
Syllabus for Instructors

Text: Peter J. Olver & Chehrzad Shakiban

Chapter 1. Linear Algebraic Systems (6 hours)
Chapter 1 should be covered rapidly but carefully since all students should have some experience with linear systems and determinants from calculus. Gaussian elimination will be used throughout the course to find the four fundamental matrix subspaces (introduced inchapter 2). Determinants will be crucial to determine eigenvalues and students should become familiar with their basic properties.
1.1. Solution of Linear Systems (.5 hr)
1.2. Matrices and Vectors (.5 hr)
1.3. Gaussian Elimination - Regular Case (1 hr)
1.4. Pivoting and Permutations (1 hr)
1.5. Matrix Inverses(omit Gauss-Jordan Elimination) (1 hr)
1.6. Transposes and Symmetric Matrices (.5 hr)
1.8. General Linear Systems (1 hr)
1.9. Determinants (.5 hr)

Chapter 2. Vector Spaces and Bases (6 hours)
This chapter should lead to (and give a proof of) the fact that a vector space has a well-defined dimension (Theorem 2.29 in section 2.4). It also gives a first strong application of the concepts just developed (i.e. the fundamental matrix subspaces) to graphs and incidence matrices. The application section 2.6. sets the stage for electrical networks, which will be covered in section 6.2.
2.1. Real Vector Spaces
2.2. Subspaces
2.3. Span and Linear Independence
2.4. Bases and Dimension
2.5. The Fundamental Matrix Subspaces
2.6. Graphs and Incidence Matrices

Chapter 3. Inner Products and Norms (4 hours)
Inner products other than the dot product are used in the applications in chapter 5 and 6 and are essential for the proof of the spectral theorem. If one plans to prove the spectral theorem via hermitian inner products, these should be introduced here (see comments on use of complex numbers in chapter 8 below). If the spectral theorem is to be proved using minimization of quadratic forms, then the Cauchy-Schwarz inequality is important, in particular that it continues to hold for positive semi-definite symmetric bilinear forms (without the "strict inequality" part of the statement of Cauchy-Schwarz).
3.1. Inner Products
3.2. Inequalities
3.4. Positive Definite Matrices
3.5. Completing theSquare (first subsection only)

Chapter 4. Minimization and Least Squares Approximation (3 hours)
The main sections are 4.3. (least squares and the closest point) and section 4.4., the application to data fitting. Least squares and their applications will be taken up again in greater depth in section 5.5.
4.1. Minimization Problems
4.2. Minimization of Quadratic Functions
4.3. Least Squares and the Closest Point
4.4. Data Fitting and Interpolation (first subsection only)

Chapter 5. Orthogonality (6 hours)
The applications in the second part of section 5.5. (in particular example 5.42 and its introduction) provide a good test for the students to see if the abstract concepts (subspaces, span, orthogonality, non-standard inner products) are understood. The second part of section 5.6 explains the geometry of matrix multiplication, setting the stage for chapter 7 on linear functions.
5.1. Orthogonal Bases
5.2. The Gram-Schmidt Process (first subsection only)
5.3. Orthogonal Matrices (omit Householder's Method)
5.5. Orthogonal Projections and Least Squares
5.6. Orthogonal Subspaces

Chapter 6. Equilibrium (2 hours)
Many students are familiar with the physical laws governing electrical networks (i.e. Kirchhoff's and Ohm's laws) but have not seen that these laws can be captured in terms of the geometry of matrix multiplication, using the incidence matrix of the underlying graph and the conductance matrix of the network.
6.2. Electrical Networks

Chapter 7. Linearity (3 hours)
The main section is 7.2. Students should see the basic linear transformations (rotation, reflection, stretching, and shear), learn how to represent the linear transformation by a matrix after choosing a basis, and understand how the matrix representation depends on the choice of basis. This is studied in greater depth for diagonalizable matrices in chapter 8.
7.1. Linear Functions
7.2. Linear Transformations

Chapter 8. Eigenvalues (5 hours)
Most students taking this course are already familiar with complex numbers. A brief review of how to manipulate them (addition, multiplication, conjugation, and modulus) and an appeal to the fundamental theorem of algebra suffice for the calculation of eigenvalues. A proof of the spectral theorem avoiding Hermitian matrices and complex inner products can be build on the minimization characterization of the eigenvalues of a symmetric matrix. This is alluded to in the second part of section 8.4., but the argument is not given in the text.
8.1. Simple Dynamical Systems
8.2. Eigenvalues and Eigenvectors
8.3. Eigenvector Bases and Diagonalization
8.4. Eigenvalues of Symmetric Matrices

Chapter 9. Linear Dynamical Systems (4 hours)
9.1. Basic Solution Techniques
9.2. Stability of Linear Systems
9.3. Two-Dimensional Systems

Leeway and Examinations (4 hours)

Total (43 hours)