65086 - Linear Algebra

Learning outcomes

The aim of this course is to provide a standard introduction to linear algebra and matrix analysis. By the end of the course the student should: - be familiar with basic concepts and properties of finite dimensional real vector spaces - be familiar with algebra of real matrices - be familiar with basic concepts and properties of euclidean spaces - be able to solve linear systems and to find the least square solution  - be familiar with linear transformations between standard real vector spaces  and their representation by matrices - be able to find eigenvalues and eigenvectors of   an endomorphisms or a square matrix and to diagonalize it, whether it is possible - be able to find the spectral decomposition of a symmetric matrix- be able to classify real quadratic forms - be able to find the singular value decomposition of a matrix and the pseudoinverse.

Course contents

Vectorial and euclidean structure of R^n; matrices; determinants; linear systems: exact solutions and least square solutions; linear applications;  eingenvalues and eigenvectors; similarity of matrices and diagonalizable matrices; symmetric matrices and spectral decomposition; quadratic form; singular value decomposition and pseudoinverse of a matrix and their application to the least square solution problem.

 

For the detailed program of the course (including the proofs of the theorems required for the oral exam) the student can refer to the notes of the lectures and to Lay's book section 7.4 (see Bibliography).

Readings/Bibliography

Theory:

• lecture notes posted on AMS campus
• "Linear Algebra and its applications" David C. Lay Addison-Wesley (2012), available at http://whitemyth.com/sites/default/files/downloads/UniDocs/Linear%20Algebra%20and%20Its%20Applications%204E%20%28Lay%29.pdf
• "Linear Algebra" Jim Hefferon available on-line at http://joshua.smcvt.edu/linearalgebra/book.pdf
• "Introduction to Linear Algebra" Gilbert Strang, Wellesley Cambridge Press (1998); video lectures  are available at http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/

Exercises:

• exercise sheets posted on AMS campus
• "Exercises and problems in linear algebra" J. M. Erdman http://web.pdx.edu/~erdman/LINALG/Linalg_pdf.pdf
• "Exercise and solution manual for a First Course in Linear Algebra" R. A. Beezer available at http://linear.ups.edu/download/fcla-3.20-solution-manual.pdf

Teaching methods

Blackboard and/or digital presentation.

Assessment methods

The exam consists of a written part and oral one both compulsory and on the whole program (see course contents).

The written part has the aim to test the ability of using linear tools to solve exercises and problems and lasts two hours. During the written test the students are allowed (and advised) to use books, lecture notes, calculator,...

In order to take the oral exam the score of the written test should be at least 15/30.

The oral exam has the aim to test the theoretical knowledge and the comprehension of the topic developed during the course and the ability of using correctly the mathematical formalism. It starts with a discussion about the written test. Then the student should answer in writing to at least two open questions.

It is compulsory to register on Almaesami only for the written test. To take both the written and the oral part, the student must show an identity document.

 

Teaching tools

Lecture notes on the topics developed during the lessons will be available on AMS campus.

Office hours

See the website of Alessia Cattabriga