Lineare Algebra (401-0131-00L, HS24)

Diese Seite wir fortlaufend ergänzt und ist noch nicht vollständig.

Dozenten

Sprache

Deutsch (Vorlesung, einige Übungsgruppen), English (some exercise classes). All teaching materials will be in English.

Mathe-Vorschau

Dieses Material führt informell einige grundlegende mathematische Begriffe ein, die in der Vorlesung benutzt, dort aber nicht vertieft erklärt werden. Ausserdem gibt es Ihnen einen Einblick, wie die Mathematik "funktioniert", die dann auch Inhalt der Vorlesung sein wird. Teile dieses Materials werden in der Vorlesung Diskrete Mathematik formal eingeführt und vertieft.

Einige Studierende werden das Material aus dem Gynmnasium kennen, für andere wird es neu sein. Wir empfehlen Ihnen, das Material durchzusehen und die Teile genauer anzuschauen, mit denen Sie nicht vertraut sind. Die integrierten Übungsaufgaben helfen Ihnen dabei, Ihr Wissen zu überprüfen. Ihr Browser merkt sich den bisher gemachten Fortschritt.

Die fünf Reading Assignments sind keine Pflichtlektüre; Sie werden der Vorlesung auch ohne deren Bearbeitung folgen können. Es handelt sich um ein Angebot, mit dem wir versuchen möchten, Ihnen die Arbeits- und Denkweise der (universitären) Mathematik schon einmal näherzubringen. Wir hoffen aber, dass Sie auch Freude daran haben!

Exam Info

The exam will be available in both German or English, and it will be possible to answer in either language.

You are allowed to bring a 6-page summary to the exam, the official guidelines for this (copied from VVZ) are: "6 A4-Seiten Notizen oder 3 doppelseitige A4 Notizen (mit LaTeX oder Ähnlichem erfasste und gedruckte Notizen sind erlaubt; sie sollten ohne Lupe lesbar sein); ein Wörterbuch (Deutsch-Englisch oder andere Fremdsprache); kein Taschenrechner."

Course Structure

The course is structured in weeks, starting with Week 0.
Week 0 consists of a single lecture on Wednesday, September 18.
Week 1 consists of two lectures, the one on Friday September 20, and the one on Wednesday September 25.
Week 2 consists of two lectures, the one on Friday September 27, and the lecture on Wednesday October 02.
...
In particular, every week except Week 0 consists of exactly two lectures (Fri and Wed).

Apart from lectures, the course also consists of solving written assignments and solving auto-corrected quizzes in Moodle. Both are released weekly on Wednesdays (i.e. at the end of the course week). Solutions to written assignments are released with a delay of one week. Moreover, every assignment will have one (specially marked) exercise for which you can hand in a solution via Moodle and receive feedback from your TA.

The auto-corrected quizzes will be made available through Moodle. These quizzes can be attempted infinitely often. They are programmed such that you will get the same question with different numbers in every attempt.

During lectures, we will sometimes make use of 'Clicker'-questions. These are short multiple choice questions that you can answer using the ETH EduApp (available through app stores, but there is also a web-interface here). We recommend that you have access to this during lectures (on your smartphone or any other device) in order to take part.

Finally, there are 26 exercise sessions that take place every week (see further below). You can enroll into one of the sessions via mystudies. Note that the sessions differ in language, focus, time and place. If you have questions regarding the course content or the assignments, these sessions are the best place to ask them. The first set of exercise sessions take place already on Thu 19.09.24 and Fri 20.09.24, respectively.

Bonus Tasks

In addition to the written assignments and the auto-graded quizzes, there will be one bonus task per week. By solving bonus tasks, you can collect a bonus for your final grade. The results from your 10 best bonus tasks will determine the grade bonus that you receive in the end. Every bonus task has the same weight, so it does not matter which 10 you solve. At most, you can get a bonus of +0.25 for your final grade. Every bonus task is either an auto-graded quiz or a written exercise that you will have to hand in. If the bonus task is an auto-graded quiz, you will be able to access it through Moodle. You will get 3 tries for the quiz and your best score will be counted. If the bonus task is a written exercise, you have to upload your handwritten (or Latex) solution via Moodle. A TA will then grade your solution.

You are required to solve the bonus quizzes (on Moodle) on your own. For the written bonus tasks, you need to hand-in your own independent solution (written by you), but you are allowed to discuss (verbally!) with other students that are taking the course as well.

Please note that bonus points from previous iterations of this course cannot be counted. So if you want to get a grade bonus for your exam, you need to do the bonus tasks of this year's iteration of the course.

Course Material and Assignments

The following table displays the content of the lecture organized by weeks. Every week is associated with two lectures (except Week 0), one assignment, solutions to the assignment, and a bonus task (except Week 0). In the table you also find the learning goals for every week, and you can see which sections of the lecture notes are taught in each week.

Lecture Notes: For the first half of the course, you can find the lecture notes here: part 1 . For the second half of the course, you can find the lecture notes here: part 2. Let us know if you find any mistake. We will try to keep an up-to-date list of typos/mistakes in the lecture notes.

As mentioned under course structure, both assignments and auto-corrected quizzes, as well as the bonus task, will be released weekly on Wednesdays. Solutions to assignments will be released with a delay of one week.

Some of the material below is marked as "CS-Lens" (typically, these are slides). This material highlights connections of the course content with Computer Science. Material marked as "CS-Lens" is not relevant for the exam.


Week	Dates	Sections	Lecture Plan	Notes / Slides	Assignment	Bonus	Learning Goals

0	Sep 18	1.1	plan	slides tablet	assignment solution	quiz	basic vector operations (in R^m): add two vectors, multiply a vector with a scalar, compute linear combinations of two or more vectors; visualize and understand these operations geometrically

1	Sep 20 Sep 25	1.2 & 1.3	plan	tablet	assignment solution	quiz	compute with vectors: scalar product, length, cosine formula, Cauchy-Schwarz inequality, triangle inequality, perpendicular vectors; define linear independence of vectors in three different ways; work with the span of vectors

2	Sep 27 Oct 02	2.1 & 2.2	plan	tablet extra material	assignment solution	quiz	compute with matrices: matrix-vector multiplication, column space, row space, rank; perform matrix multiplication, including matrix- vector, vector-matrix, scalar and outer product, distributivity, associativity

3	Oct 04 Oct 09	2.2 - 3.1	plan	tablet CS Lens animation	assignment solution	hand-in	explain the CR decomposition; linear transformations, visualizing linear transformations in 2d, properties of linear transformations, matrix representation of linear transformations; systems of linear equations, systems of linear equations with unique solutions

4	Oct 11 Oct 16	3.2 & 3.3	plan	tablet beautiful slides ugly slides	assignment solution	quiz	do elimination and back substitution on square systems of linear equations, explain when and why this works or fails; define the inverse of a matrix, compute inverses of 2 × 2 matrices, characterize when the inverse exists (The Inverse Theorem), invert a product of matrices and the transpose of a matrix;

5	Oct 18 Oct 23	3.4 & 3.5	plan	tablet CS Lens	assignment solution	quiz	derive and explain the LU factorization from elimination; compute the REF and RREF of a given m x n matrix A, explain why it equals R in A=CR;

6	Oct 25 Oct 30	4.1 & 4.2	plan	tablet CS Lens slides	assignment solution	hand-in	explain the concept of a vector space; give examples that are not R^m; define and identify subspaces; explain when vectors span a subspace / form a basis of it; prove that every basis has the same number of vectors; define the dimension of a vector space; find a basis for a given vector space / subspace;

7	Nov 01 Nov 06	4.3 & 5.1	plan (Friday)	tablet slides (5.1 & 5.2)	assignment solution	quiz	define the nullspace of a matrix; compute a basis for the nullspace of a matrix; solve Ax=b by elimination to REF, read off all solutions, count the number of solutions; define the four fundamental subspaces of a matrix: column space, row space, nullspace, left nullspace; compute their dimensions, depending on shape and rank of the matrix; define orthogonal complement and orthogonal subspaces; prove that nullspace and row space of a matrix are orthogonal; argue about dimensions of two orthogonal subspaces;

8	Nov 08 Nov 13	5.2 & 5.3		slides (5.1 & 5.2) slides (5.3 & 5.4) tablet	assignment solution	quiz	Define Projection, Derive formula for Projection on a subspace, Compute the Projection Matrix. Show that when A has independent columns, A^TA is invertible and symmetric. Define Least Squares solution, derive Normal equations, compute a least squares solution. Use Least Squares to fit a line to points (linear regression).

9	Nov 15 Nov 20	5.4		slides (5.3 & 5.4) slides (5.4 & 5.5) CS Lens tablet	assignment solution	hand-in	Orthogonal vectors, Orthonormal vectors, Orthonormal basies. Orthogonal Matrices. Orthogonal matrices preserve norm and inner-product. Projections with orthonormal bases. Build an orthonormal basis with Gram-Schmidt (and show correctness of Gram-Schmidt). QR decomposition. Projections and least squares with QR decomposition.

10	Nov 22 Nov 27	5.5 & 5.6		slides (5.4 & 5.5) slides (5.6)	assignment solution	quiz	Pseudo-inverse, definition and properties. Pseudo-inverse and minimum norm solution. Pseudo-inverse and projection. Polyhedron, projections of sets, Farkas lemma.

11	Nov 29 Dec 04	6 & 7.0		slides (6) slides (7.0 & 7.1) CS Lens	assignment	quiz	Determinant and its properties, definition via permutations, connection to matrix inverse, co-factors and the determinant, Cramer's rule. Complex numbers, calculations with complex numbers. Fundamental theorem of algebra, roots of polynomials. Complex-valued vectors and matrices. Eigenvalues and eigenvectors, definition and 2x2 examples.

12	Dec 06 Dec 11	7.1 & 7.2		slides (7.0 & 7.1) slides (7.2 & 7.3)			Characteristic polynomial, algebraic multiplicity, finding eigenvalues and eigenvectors, properties of eigenvalues and eigenvectors. Linear independence of eigenvectors corresponding to distinct eigenvalues. Determinant, trace, and connection to eigenvalues. Eigenvalues and eigenvectors of rotations and other linear transformations. Eigenvalues and eigenvectors of orthogonal matrices. Eigenvalues and eigenvectors of diagonal matrices. Eigenvalues and eigenvectors of projection matrices. Repeated eigenvalues and geometric multiplicity. Linear independence of eigenvectors, complete sets of real eigenvectors. Change of basis, diagonalization, diagonalizable matrices. Similar matrices, eigenvalues of similar matrices.

13	Dec 13 Dec 18	7.3 & 8

Exercise Sessions

There are 26 different exercise sessions that take place weekly. One of them is held online, all other sessions take place in-person. Some of the sessions are held in German, others in English, and there is one session in Italian. Moreover, some of the sessions are marked as 'focus group'. Focus groups are intended for students who describe their own prior knowledge as below average. The focus group TAs therefore focus on the relevant basics and explain concepts as well as exercises in more detail. Apart from this, the focus groups cover the same content as the regular groups. Please sign up for one of the sessions in mystudies.


G-01	English		Thu 8-10am	CAB G 56	Tanguy Magne

G-02	English		Thu 8-10am	CHN C 14	Sofia Giampietro

G-03	German		Thu 8-10am	CHN D 42	Mattia Taiana

G-04	German		Thu 8-10am	CHN D 46	Silvan Bolt

G-05	English	Focus Group	Thu 8-10am	CLA E 4	Bogdan Murgu

G-06	German		Thu 8-10am	HG F 26.5	Simon-Philipp Merz

G-07	English		Thu 8-10am	IFW A 32.1	Metehan Kiliç

G-08	Italian		Thu 8-10am	ML F 34	Matthieu Croci

G-09	German	Focus group	Thu 8-10am	RZ F 21	Hannah Schwede

G-10	German	Focus Group	Thu 8-10am	HG E 22	Till Schnabel

G-11	German		Thu 8-10am	ML F 38	Anna Mihalkova

G-12	German		Thu 8-10am	ML H 41.1	Elia Trachsel

G-13	English		Thu 4-6pm	NO C 6	Jing Ren

G-14	English		Thu 4-6pm	IFW A 34	Xiang Zhang

G-15	English		Thu 4-6pm	CHN D 42	Julio Cabello

G-16	German	Focus Group	Thu 4-6pm	CHN D 44	Stefan Kuhlmann

G-17	German		Thu 4-6pm	ML J 34	Tae Kim

G-18	English		Thu 4-6pm	LFW C 5	Gabriele Fadini

G-19	English	Focus Group	Thu 4-6pm	IFW B 42	Mohamed-Hicham Leghettas

G-20	German		Fri 2-4pm	IFW A 34	Mark Sosman

G-21	English		Fri 2-4pm	CHN D 42	Oleksandr Kulkov

G-22	English		Fri 2-4pm	ETZ E 9	Aviv Segall

G-23	English	Focus Group	Fri 2-4pm	CAB G 51	Manuel Di Sabatino

G-24	German		Fri 2-4pm	CHN G 46	Benjamin Gruzman

G-25	German	Focus Group	Fri 2-4pm	IFW C 31	Sarina Michael

G-26	English		Fri 2-4pm	online	Saleh Ashkboos

Theory of Combinatorial Algorithms