Ing Ind - Inf (Mag.)(ord. 270) - CO (482) COMPUTER SCIENCE AND ENGINEERING - INGEGNERIA INFORMATICA
088975 - ADVANCED LINEAR ALGEBRA
The aim of this course is to provide the basic mathematical tools necessary for a Master Degree in Engineering, with a particular emphasis to Linear Algebra. The course covers the mathematical background of the most common techniques for treating linear engineering models from least squares to Gauss-Newton method and SVD. Each topic is treated not only theoretically but also practically through lab sessions that allow students to acquire a good level of autonomy in problem solving.
Risultati di apprendimento attesi
Expected learning outcomes
Knowledge and understanding
Understand the geometry of euclidean spaces and the associated constructions: orthogonal decomposition, Gram-Schmidt procedure, QR-decomposition
Understand the most important ideas of linear algebra from the notion of space to diagonalization and other matrix decompositions and constructions.
Recognize the linearity of many engineering problems and exploit it by applying to them the ideas of linear algebra
Understand matrix norm, minimal gain, singular values, condition number and their significance in handling errors in measures and designs
Applying knowledge and understanding
Know how to compute explicitly the most common matrix decompositions: QR, SVD, diagonalization
Extract the matrices of a linear model from input and output data
Determine when least squares is applicable and find least squares solutions
Handle multiobjective least squares problems
Find least norm solutions for underdetermined systems
Know how to apply Gauss-Newton method to nonlinear models
Know how to compute matrix norm, minimal gain, and the exponential of a diagonalizable matrix
Apply SVD to solve problems with constraints
Apply matrix exponential and diagonalization in the qualitative study of a linear autonomous systems
The following topics are covered:
Review of basic linear algebra: vector spaces, subspaces. linear combination, linear independence, basis.
More linear algebra: scalar product, orthogonality, orthonormal basis, Gram-Schmidt orthonormalization, QR factorization.
Least-squares procedure and applications.
Regularized least-squares and Gauss-Newton method
Least-norm solutions of underdetermined equations
Eigenvectors and diagonalization
Symmetric matrices, quadratic forms, matrix norm
Singular Value Decomposition
Matrix exponential and autonomous systems
Jordan canonical form and Cayley-Hamilton theorem
Labs will be offered with detailed solutions of engineering problems using the material of the course and matlab.
The required background is limited to standard introductory courses in Calculus and Linear Algebra.
Modalità di valutazione
On course evaluation: There are a midterm and a final exam. Also lab reports will be evaluated. The midterm exam weights 30% of the final mark. The final exam weights 50% of the final mark. The lab reports weight 20% of the final mark. There will be two labs. Attendance to the labs is mandatory for acquiring the lab score.
The midterm consists of a homework on the first part of the course.
The final exam is a written examination on the syllabus of the second part of the course.
The final exam can be taken in any exam session and its score can be rejected retaining the lab and midterm scores.
Off course evaluation: a written examination on the whole syllabus including the material discussed in lectures, recitations, and labs.
Type of assessment
Solution of numerical problems
Solution of engineering models developped in class
Assessment of laboratorial artefacts
Solution of engineering models assigned as homework (individual) or as matlab assignment (in groups)