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Math 415. Appled Linear Algebra
Syllabus for Instructors
Students in this course are primarily from engineering and related departments
such as physics. Emphasis should be on explaining and relating basic concepts
of linear algebra, and showing how they apply to, and arise in, concrete
problems. In general, simple proofs should be included only as time permits,
and outlining the key arguments of a proof in a special case often suffices.
However, students do need to be able to do and understand manipulations
in abstract vector spaces(such as testing for linear independence).
Applications to be covered are in sections 2.6., 4.4., 5.5., 6.2., and chapter 9.
Note: The overall pace in this course will be demanding, so plan carefully.
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 throughoutthecourseto findthefourfundamental
matrixsubspaces(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
1.2. Matrices and Vectors
1.3. Gaussian Elimination—Regular Case
1.4. Pivoting and Permutations
1.5. Matrix Inverses(omit Gauss-Jordan Elimination)
1.6. Transposes and Symmetric Matrices
1.8. General Linear Systems
1.9. Determinants
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 theconcepts justd eveloped(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 andOhm’slaws) 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)
| Department
of Mathematics 273 Altgeld Hall, MC-382 1409 W. Green Street, Urbana, IL 61801 USA Telephone: (217) 333-3350 Fax: (217) 333-9576 Email: office@math.uiuc.edu |
College of Liberal Arts and
Sciences Last modified November 8, 2007 |