Instructor: **Paweł Pilarczyk**

Teaching Assistant: **Kristóf Huszár**

Course dates: October 6 - November 19, 2014.

At the origin of Linear Algebra lies the study of systems of linear equations and the techniques to solve them. These appear all over science, in particular in the analysis of scientific data. Moreover, Linear Algebra is fundamental in much of modern mathematics and is, in itself, a good example of what a mathematical theory is and how mathematics works.

The goal of this course is twofold. First is to introduce the most important concepts and techniques of Linear Algebra: vectors and vector spaces, matrices, linear functions, eigenvalues and eigenvectors, and how to solve linear systems of equations. Second is to expose students to the way mathematicians think and work; in other words, to the specific mathematical language and logic that is present in the formulation of mathematical definitions and statements.

The lecture will essentially follow the material taught at the previous editions of the Linear Algebra course at IST, with small changes and adjustments. The Web pages of the previous courses are available online at:

- https://pub.ist.ac.at/courses/2012/linearalgebra/
- https://pub.ist.ac.at/courses/AY1314/LinearAlgebra_F13/

For a succint list of topics actually covered in each lecture, please, see the schedule below.

The student successfully completing the course should be able to:

**decide**whether specific statements are true or false and**justify**this,**state**definitions introduced in the course and**verify**whether specific objects satisfy them or not,**recall**theorems given in the course and**explain**them,**prove**those statements that were proved in the lecture, and also similar ones,**identify**linear algebra problems in various contexts (e.g., computer science or data analysis),**apply**the covered methods and theorems to solve problems, including but not limited to: solving linear equations, changing bases, inverting matrices, finding eigenvalues and eigenvectors

The final grade consists of:

- 30% homework,
- 15% activity during the lectures,
- 15% activity during the recitation sessions,
- 40% final exam.

Homework can be done individually or in groups of up to 3 people. Solutions will have to be handed by the date that is given on the assignment sheet (typically within a week from the assignment date). Solutions should be hand written (no printouts or xeroxed copies, please). If solving in a group, all the students must be able to explain all the solutions, and each student in the group must write him or herself at least some of the solutions.

Activity during the lectures will consist of solving short quizzes that will be given at the beginning or in the middle of some of the lectures and participation in discussions during the interactive part of the lectures.

Recitation classes will be entirely interactive, and students' activity will consist of discussions and presentations.

The final exam will be given on Wednesday, November 19, in the morning (note the exact exam time and the classroom different from Mondi 3). Students will be given a set of problems to solve individually, without using any additional materials (books, calculators, notes, phones, computers, etc.) The exam problems will be inspired by homework and by the quizzes and discussions in the lectures and also during the recitation classes.

No. | Date | Lecture | Recitation | Topics Covered in the Lecture |
---|---|---|---|---|

1 | Mon, Oct 6 | 13:45-15:00 | 15:15-16:00 | mathematical logic; definition of a vector space |

2 | Wed, Oct 8 | 13:45-15:00 | 15:15-16:00* | linear maps, matrices, matrix-vector multiplication; linear regression |

3 | Mon, Oct 13 | 13:45-15:00 | 15:15-16:00 | linear functions from R^{n} to R^{m}; matrix-matrix multiplication |

4 | Wed, Oct 15 | 13:45-15:00 | 15:15-16:00* | data matrices; linear (in)dependence of vectors |

5 | Thu, Oct 16 | 8:45-10:00 | bases; dimension | |

6 | Mon, Oct 20 | 13:45-15:00 | 15:15-16:00 | Gaussian elimination method; invertibility |

7 | Wed, Oct 22 | 13:45-15:00 | inner product, norm, metric; matrix transpose | |

8 | Mon, Oct 27 | 13:45-15:00 | 15:15-16:00 | normalized vectors, orthogonal matrices, projections; connection to statistics |

9 | Wed, Oct 29 | 13:45-15:00 | 15:15-16:00* | orthogonal bases; eigenvalues and eigenvectors |

10 | Mon, Nov 3 | 13:45-15:00 | 15:15-16:00 | Markov chains; eigenvalues in the analysis of dynamical systems |

11 | Wed, Nov 5 | 13:45-15:00 | 15:15-16:00* | matrix diagonalization; principal component analysis |

(Nov 10 and 12) | (no classes) | |||

12 | Mon, Nov 17 | 13:45-15:00 | 15:15-16:00 | selected interesting applications of linear algebra |

Wed, Nov 19 | 9:45-11:00 |
FINAL EXAM | Location: in the Computer Science room (Central Bldg, 2nd Floor) |

During the last lecture, on November 17, 2014, we are going to discuss selected applications of linear algebra, chosen from the book "Thirty-three Miniatures" by Jiri Matousek. The book can be borrowed from the IST library, or downloaded from author's webpage. The students are invited to give presentations based on a section of their choice. The procedure will be as follows:

- Get a copy of the book, either from the library, or from the Web.
- Read the table of contents, take a glimpse of the contents, read some parts.
- Choose a section for the presentation, something that sounds interesting for you and is adequate for your level of knowledge. Make sure that each person has chosen a different section from the book. Communicate your choice to Kristóf. He will put the information at the course website.
- Prepare a 10-15 minute-long presentation. Don't prepare slides, write everything on the blackboard. You can only use the computer to show additional material (photos, videos, etc.) if relevant (but this is optional). Focus on convincing the audience that the application you chose is actually interesting.
- Come to the lecture on November 17 and deliver your presentation. Be prepared to answer questions and explain the material presented.

If you need any help with the book, Kristóf will be more than happy to assist you. He can help you choose a section appropriate for you, and can explain some difficult parts of the material, should this be necessary. Please, note that both Kristóf and Paweł will be absent during the week Nov 10-14, so you can count on our help before that week, and then work on your own.

P.S. If you prefer to give a presentation about another interesting application of linear algebra that is not in the book then this will also be good. Please, communicate the title and a short description to Kristóf and get his permission.

Notes for selected lectures: lecture2, lecture3, lecture4-5, lecture6, lecture7, lecture8, lecture9, lecture10, lecture11.

Homework assignments: homework1, homework2, homework3, homework4, homework5, homework6.

Please note that you must log into the IST Intranet to download the files.