Course Description:
This is a course aiming for undergraduate students majoring in math and engineering who are interested in understanding the importance of linear algebra and its application to problems within and outside mathematics. The basic concepts and results of linear algebra (vector spaces, linear transformations, matrices, determinants, eigenvalues and eigenvectors, orthogonality and diagonalization) will be reviewed. The course will cover the following topics: least square approximation, discrete Fourier transformation, numerical computations with matrices, graphs and networks, and image compression. Possible additional topics include: finite element method, systems of ordinary differential equations and linear programming.
The instructor may set MatLab assignments at their discretion.
Prerequisites:
Math 250 Introductory Linear Algebra and Math 251 Multivariable Calculus.
Textbook:
For current textbook please refer to our Master Textbook List page
Published version of the textbook available in Spring 2016 from World Scientific Publishing
Other Resources
- Survey of course topics by Roe Goodman
Discrete Fourier and Wavelet Transforms: Mathematical Microscopes for Signal Processing - Fast Fourier Transform links
- Wikipedia page on Wavelets
- An article on Image Compression and the JPEG 2000 algorithm based on the CDF Wavelet transform (which is studied in this course).
- An article on Discrete Wavelet Transformations and Undergraduate Education by C. Beneteau and P. J. Van Fleet (from Notices of the American Mathematical Society, May 2011) that outlines all the mathematical topics covered in the course with many interesting examples of image processing.
- The MIT Open CourseWare page of Gibert Strang's course Wavelets and Filter Banks.
- Wavelet books and link
Other Recommended Books (not required for course)
A. Jensen and A. la Cour-Harbo, Ripples in Mathematics: The Discrete Wavelet Transform
S. Allen Broughton and Kurt Bryan, Discrete Fourier Analysis and Wavelets
James S. Walker, A Primer on Wavelets and Their Scientific Applications (Second Edition)
Topics
Geometry of Linear Equations, Gaussian Elimination
Matrix Multiplication, LU Decomposition
Row Operations, Inverses and Transposes
Vector Spaces and Subspaces, Kernel and Range of Matrices
Linear Independence, Basis, Dimension
Solving Ax=b
Graphs and Networks
Linear Transformations
Orthogonal Vectors and Subspaces
Orthogonal Projection
Least Square Method
Gram-Schmidt Procedure, QR Decomposition
Fast Fourier Transform
Determinants and Their Properties
Applications of Determinants, Eigenvalues
Diagonalization of Matrices
Matrix Powers and Difference Equations
Matrix Exponentials and Differential Equations, Similarity Transformations
Minima, Maxima, and Saddle Points
Positive Definite Matrices
Singular Value Decomposition
Minimum Principles, Finite Element Method
Matrix Norms and Condition Number
Iterative Methods to Solve Ax=b
Sample Course Materials
Sample MATLAB Assignments
- Project 1: Digital Signals and Vector Graphics (pdf format)
- Project 2: Convolution and Discrete Fourier Transform (pdf format)
- Project 3: Haar Wavelet Transform (pdf format)
- Project 4: Implementation of Wavelet Transforms (pdf format)
- Project 5: Image Analysis by Wavelet Transforms (pdf format)
For Project 2 you will use the Finite Fourier transform graphic user interface.
Matlab m-file. Here is the link to download this m-file: fftgui
For Projects 4 and 5 you will use the Uvi_Wave collection of Matlab m-files for wavelet transforms (developed at the University of Vigo, Spain). Here is the link to download these m-files: Uvi_Wave zip file (unzip the file to use the package)
Using Matlab
Note: You can run Matlab on your own computer (without buying the program) by using the Rutgers X-application server.
- Click on this apps server link.
- Log in to the apps server using the connect button at the upper right-hand corner of the screen and your Rutgers NetID.
- From the Main Menu at the lower left corner of the apps server toolbar, click on Education and then on Matlab
- From the Main Menu click on Internet and then on Firefox Web Browser to access the Uvi_Wave files from the math 357 course web page.
- Copy the fftgui.m file and the whole unzipped Uvi_Wave directory into a directory that your create on the X-apps server. Then set the Matlab path to this directory.
Course History
Taught by Prof. R. Goodman 2005-2008 and 2010-2014, Prof. V. Retakh 2009 and 2016, Dr. M. Thibault 2015.
Schedule of Sections:
01:640:357 Schedule of Sections