35212 Computational Linear Algebra

Warning: The information on this page is indicative. The subject outline for a particular semester, location and mode of offering is the authoritative source of all information about the subject for that offering. Required texts, recommended texts and references in particular are likely to change. Students will be provided with a subject outline once they enrol in the subject.

Handbook description

In this subject, students develop familiarity with the theory of finite dimensional linear algebra, applications of this theory in areas such as statistical analysis and the solution of differential equations and some of the main computational techniques used in these applications. Topics include systems of linear equations (LU factorisation and iterative methods); vector spaces; inner product spaces; Gram-Schmidt orthogonalisation, QR decomposition; approximation theory: least squares and orthogonal polynomials; the eigenvalue problem; Singular value decomposition and applications.

Subject objectives/outcomes

Students who successfully complete this subject should:

1. have developed and be able to demonstrate skills in theoretical and computational techniques of linear algebra ;
2. understand and be able to explain the principal ideas and results which justify the above techniques ;
3. be able to conduct rigorous mathematical reasoning for the problems relevant to the subject and be able to understand mathematical proofs presented in the subject;
4. be able to communicate clearly in the mathematical terminology of linear algebra;
5. be able to implement computational algorithms in linear algebra problems using Mathematica.

Contribution to course aims and graduate attributes

The ideas and techniques introduced in this subject are further developed and applied in a wide range of other subjects in the areas of differential equations, mathematical methods, optimisation and statistics, all of which underpin the professional and research practice of mathematical techniques. The subject is taught with a strong numerical emphasis and thus the laboratories are an important component of the coursework, serving as an introduction to the routine use of numerical approaches for solving mathematical problems, and also as a tool to develop further the use of integrated computational systems such as Mathematica (and computer programming) in professional mathematical practice.

The subject contributes to strengthening the attributes of graduates in many areas. Students' personal skills in managing their own work are enhanced, as are: their awareness of the importance of self-motivation and time-management; their ability to identify personal goals and accept responsibility for achieving them; their acceptance of responsibility for their own decisions; their capacity for initiative and innovation. Students also broaden their disciplinary, professional and technical knowledge, and their ability to define and solve problems. Their intellectual development is strengthened in the areas of:

• critical and independent thinking; spoken and written communication; academic literacy;
• information technology literacy; information literacy.

Teaching and learning strategies

Weekly on campus: Three 1 hr lectures, one 1 hr laboratory Face-to-face classes will incorporate a range of teaching and learning strategies including short presentations, discussion of readings and student groupwork. Classes each week should be supported by at least five hours per week of individual or group study, developing and practising skills by doing many textbook questions etc.

Content

The course content will be motivated by a number of paradigm applications. These include the numerical solution of partial differential equations, principal component analysis from statistics, and the use of transfer matrix methods in the physical sciences. Review of row reduction methods for solving linear systems, including a discussion of round off errors and their avoidance with pivoting strategies. LU factorization techniques (Crout, Doolittle, Cholesky) including computational implementations, their relationship to Gaussian elimination, and their use in solving linear systems and matrix inverses. Iterative techniques for solving linear equations (Jacobi, gauss-Seidel, successive over-relaxation) and a sketch of the theory of their convergence properties based on fixed point analysis.

Vector spaces and linear transformations including discussion of null spaces, column spaces, bases, rank, and change of basis. Inner products and orthogonality, including the Gram-Schmidt process and the QR decomposition. Orthogonal polynomials and their application in approximation theory. The eigenvalue problem, consideration of methods for finding eigenvalues (power method, the basics of the QR method), matrix digitalization. Normal matrices and unitary matrices. The singular value decomposition and applications.

Assessment

Assessment Item 1: Fortnightly laboratory / hand-in tutorial work

Assessment Item 2: Individual assignment

Assessment Item 3: Class Test

Assessment Item 4: Final examination

Minimum requirements

In order to pass this subject, a student must achieve a final result of 50% or more.

Any assessment task worth 40% or more requires the student to gain at least 40% of the mark for that task. If 40% is not reached, an X grade fail may be awarded for the subject, irrespective of an overall mark greater than 50.

The final result is simply the sum of all the marks gained in each piece of assessment, with the exception that if the examination mark as a percentage is greater than the combined mark then the examination percentage will be awarded as the final mark.

Recommended texts

Lay, D. C. Linear Algebra and Its Applications, 4th Edition, Pearson, 2012.
Craddock M., and Langtry, T. N. Notes on Computational Techniques in
Linear Algebra, UTS, 2008(9). [Available via UTSOnline.]

References

• Anton, H. & Rorres, C. Elementary Linear Algebra (Applications Version), 10th Edition, John Wiley & Sons, 2010.
• Strang, G. Linear Algebra and its Applications, Harcourt Brace Jovanovich.
• Johnson, E. Linear Algebra with Mathematica, Brooks Cole, 1995.
• S.S. Rao. Applied Numerical Methods for Scientists and Engineers, Prentice Hall, 2002. ISBN: 0-13-089480-X
• W. Cheney and D. Kincaid. Numerical Methods and Computing, 6th Edition. Brooks-Cole 2007.
• G. M. Phillips and P. J. Taylor, Theory and Applications of Numerical Analysis, 2 nd edition, Academic Press, 1996.

Note that these are just a selection of useful references: there are many more texts on linear algebra and on numerical analysis available from the University Library.

Other resources

A weekly schedule for the subject, including dates of assessment tasks, will be distributed separately.