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Kursusnavn (dansk):Mathematical Analysis and Linear Algebra, MSc 
Kursusnavn (engelsk):Mathematical Analysis and Linear Algebra, MSc 
Semester:Forår 2017 
Udbydes under:cand.it., softwareudvikling og -teknologi (sdt) 
Omfang i ECTS:7,50 
Kursussprog:Engelsk 
Kursushjemmeside:https://learnit.itu.dk 
Min. antal deltagere:15 
Forventet antal deltagere:
Maks. antal deltagere:30 
Formelle forudsætninger:
Læringsmål:After the course the student should be able to:

- Use nummerical methods such as Newton’s method and Taylor series, and explain the theory behind them.
- Reflect on the applicability of a given numerical method to a given problem.
- Compute the derivatives of curves and surfaces, explain the geometric interpretation of these, and apply them to solving a wide range of mathematical problems including optimisation problems.
- Compute double integrals and explain their geometric interpretation.
- Compute the essential constructions of basic linear algebra including inverse matrices, eigenvectors and eigenvalues, matrix diagonalisation, the dimensions and bases of the four fundamental subspaces and determinants of square matrices, and explain the theory behind these constructions.
- Explain the use of linear algebra in applications such as solutions to linear equations, the method of least squares and the theory of the PageRank algorithm. Discuss the relevance of the theoretical results proved in the course for these applications. 
Fagligt indhold:This is a course in mathematics covering linear algebra and analysis of functions in many variables. These are perhaps the two areas of mathematics that have found most uses in practical applications.

Linear algebra is essentially the study of matrices, i.e., rectangular arrays of numbers. Matrices are used to represent many different things, including systems of linear equations, transformations of space (such as rotations or reflections as used in computer graphics) and probabilistic systems. Likewise, many of the internal operations in search engines are essentially operations on matrices. By studying the mathematical theory of matrices we will learn tools that are useful for all of these application areas. As an example application we look at the PageRank algorithm used in Google's search machines to rank webpages for importance.

Mathematical analysis (or calculus) is the study of the derivative and the integral of real valued functions as taught in high school, but in this course we extend the theory to functions of many variables returning vectors of real numbers. Just like the single variable case, one of the main applications is to solve optimization problems, which in real life are of many variables. This is the theory underlying many methods in artificial intelligence, but it is also important in many other areas including image analysis, business intelligence, physics and economics.

While the topics taught have been chosen for their applicability, the focus of the course will be on the mathematics. The aim is to understand the theory well enough that we can understand why the methods in the application areas work, and even use the mathematical theory to adapt existing methods or create new ones. In other words, the aim is not that the student should simply be able to use methods and insert into formulas, but rather explain why the methods work and reflect on the theory presented. For this reason the course has an oral exam. 
Læringsaktiviteter:14 ugers undervisning bestående af forelæsninger og øvelser

14 weeks of teaching consisting of lectures and exercises. The exercise sessions will train the students oral presentation skills, and so students are expected to prepare for the exercise sessions. The mandatory exercises will
mainly ask the students for written solutions to theoretical exercises, but will also include a few implementation exercises. 

Obligatoriske aktivititer:There are 6 mandatory assignments, out of which 5 must be approved for the student to qualify for the exam.
The deadlines are evenly distributed over the semester (approximately one every 2 weeks), exact dates will be posted on learnit the first week of the semester.
If a mandatory assignment is not approved the first time, the student will be allowed to resubmit exactly one week after the first deadline.
If this also fails, one more resubmission the following week is allowed.

Be aware: The student will receive the grade NA (not approved) at the ordinary exam, if the mandatory activities are not approved and the student will use an exam attempt 
Eksamensform og -beskrivelse:B1I Mundtlig eksamen med forberedelse. På ITU, (7-scale, external exam)

Oral examination with time for preparation at the exam.
Duration: 30 minutes preparation at ITU and 30 minutes oral examination.
The student draws a question and has 30 minutes preparation before 30 minutes oral examination.  

Litteratur udover forskningsartikler:- Gilbert Strang: Introduction to Linear algebra, 4th edition

- David Guichard: Multivariable calculus, early transcendentals

- Kurt Bryan and Tanya Leise: The $25,000,000,000 eigenvector, the linear algebra behind Google.