# 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 167: Applied Linear Algebra

Approved: 2018-05-30, N. Saito, J. De Loera, E. G. Puckett
Suggested Textbook: (actual textbook varies by instructor; check your instructor)
Lars Eldén: Matrix Methods in Data Mining and Pattern Recognition, SIAM, 2007
https://doi.org/10.1137/1.9780898718867
Search by ISBN on Amazon: 978-0-898716-26-9
Prerequisites:
MAT 022A or MAT 027A or MAT 067 or BIS 027A.
Suggested Schedule:

The lecture notes prepared and used by Naoki Saito are available at: https://www.math.ucdavis.edu/~saito/courses/167/lectures.html. The corresponding homework problems as well as sample midterm and final exam problems are available upon request.

 Lectures Sections Comments/Topics 1st Week Chap. 1 and from various sources (see note below); Chapter numbers refer to the book of Elden while it is easy to see the corresponding lecture slides by Saito Motivational introduction with examples; Review of basic linear algebra: meaning of matrix-vector/matrix-matrix multiplications 2nd Week Chap. 2 Review of basic linear algebra: range; null space; linear independence; bases; dimensions; ranks; inverse matrices; inner products; vector and matrix norms 3rd Week Sec. 3.6; Chap.4; Chap. 5 Introductory least squares problems; orthogonality; projectors; QR factorization; classical Gram-Schmidt orthogonalization 4th Week Chap. 5 Modified Gram-Schmidt; Householder triangularization; Givens rotations 5th Week Midterm Exam; Chap. 6 Introduction to Singular Value Decomposition (SVD) 6th Week Chap. 6, Chap. 7 Low rank approximation; condition numbers; SVD vs Principal Component Analysis 7th Week Chap. 9 Data clustering; Nonnegative Matrix Factorization 8th Week Chap. 10 Applications of SVD I: pattern classification 9th Week Chap. 11 Applications of SVD II: text mining 10th Week Chap. 12 as well as extra material, handouts Applications of SVD III: search engines (see notes below)
• In the 1st Week, it is important to motivate the students using important examples to show how linear algebra is used in real world. Suggested examples are: music signal representation and compression, image compression, web search engines, inverse problems (e.g., tomography), etc.
• In the 8th and 9th Weeks, the instructor should supply some applications of SVD including web search engines, least squares problems, image approximations, pattern classification, inverse problems in a more detailed manner some of which were discussed in the 1st Week.
• In the 10th Week, the instructor can freely choose various applications of interest. The instructor is encouraged to check the following optional textbooks for further examples and applications:
• Carl D. Meyer: Matrix Analysis and Applied Linear Algebra, SIAM, 2000.
• Lloyd. N. Trefethen and Davis Bau, III: Numerical Linear Algebra, SIAM, 1997.
• Michael W. Berry and Murray Browne: Understanding Search Engines: Mathematical Modeling and Text Retrieval, 2nd Ed., SIAM, 2005. Available free online via UCD’s SIAM subscription at https://doi.org/10.1137/1.9780898718164
• David Skillicorn: Understanding Complex Datasets: Data Mining with Matrix Decompositions, Chapman & Hall/CRC, 2007. Available free online via UCD’s subscription at https://www.taylorfrancis.com/books/9781584888338
• Cleve Moler: Numerical Computing with MATLAB. Available for free online at https://www.mathworks.com/moler/chapters.html. Chapter 2: Linear Equations, Chapter 5: Least Squares, and Chapter 10: Eigenvalues and Singular Values are quite relevant for this course.