IT-Universitetet i København
 
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Kursusbeskrivelse
Kursusnavn (dansk):Mathematical Analysis and Linear Algebra 
Kursusnavn (engelsk):Mathematical Analysis and Linear Algebra 
Semester:Forår 2014 
Udbydes under:cand.it., softwareudvikling og -teknologi (sdt) 
Omfang i ECTS:7,50 
Kursussprog:Engelsk 
Kursushjemmeside:https://learnit.itu.dk 
Min. antal deltagere:12 
Forventet antal deltagere:70 
Maks. antal deltagere:80 
Formelle forudsætninger:None. 
Læringsmål:After the course the student should be able to:
• Use Newton’s method and explain the theory behind it.
• Define and calculate partial derivatives, gradients, directional derivatives of functions in several variables.
• Calculate minima and maxima for given functions possibly under given constraints, and explain the theory underlying the methods used this.
• Argue for convergence or divergence of a given infinite series.
• Describe Taylor series and calculate a Taylor expansion for a given function. Judge if the expansion converges.
• Compute multiple integrals, arc lengths, line integrals and formulas for tangent surfaces.
• Define the concept of vector field and describe properties of specific vector fields, e.g. judge if a given vector field is conservative or source free.
• Define and give a geometric interpretation of the concepts of vector spaces and subspaces. Define and compute dot and cross products of vectors.
• Solve systems of linear equations in several variables using row operations, inverses of matrices and least squares and determine possible number of solutions for a given system of linear equations.
• Define linear transformations, symmetric, quadratic, positive definite and similar matrices, singular value decomposition, elimination, eigenvalues and eigenvectors, orthogonality, orthonomality, and explain the importance of these concepts within linear algebra.
• Describe and calculate inverses, determinants, transposes of matrices 
Fagligt indhold:The course consists of two parts. In the first we study calculus of functions of several variables. Topics include
• A brief review of calculus of functions in one variable, including differentiation, integration, the fundamental theorem of calculus and Newton’s method
• Infinite series, the concept of convergence and techniques for proving convergence or divergence of series.
• A review of dot products and cross products of vectors.
• Curves in the plane and three dimensional space
• Partial derivatives of functions in many variables, directional derivatives and gradients, the chain rule in many dimensions. Finding maxima and minima under constraints.
• Multiple integrals and Fubini’s theorem.
• If time permits we will also cover vector calculus including line integrals and Green’s theorem, which is a generalization of the fundamental theorem of calculus to many variables.
The second part is about linear algebra and its applications.
Linear algebra and matrices is perhaps the branch of mathematics that has found most uses in practical applications. Matrices form the base for many models in economics, computer vision, graphics, games and various optimization problems.
Topics include:
• Fundamental properties of linear algebra, matrices and their relationship to linear functions.
• Methods for solving linear sets of equations.
• Methods for decomposing matrices and analyzing sets of linear equations
• Practical applications of matrices 
Læringsaktiviteter:14 ugers undervisning bestående af forelæsninger og øvelser

TBA 

Obligatoriske aktivititer:There are mandatory assigments. 
Eksamensform og -beskrivelse:X. experimental examination form (7-scale; external exam), 7-trins-skala, Ekstern censur

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: Calculus. Available for free download at
http://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/
- Gilbert Strang: Introduction to Linear algebra, 4th edition

Mere info om den sidste er her
http://www.amazon.co.uk/Introduction-Linear-Algebra-Gilbert-Strang/dp/0980232716 
 
Afholdelse (tid og sted)
Kurset afholdes på følgende tid og sted:
UgedagTidspunktForelæsning/ØvelserStedLokale
Torsdag 12.00-13.50 Forelæsning ITU 2A52
Torsdag 14.00-15.50 Øvelser ITU 2A52

Eksamen afholdes på følgende tid og sted:
EksamensdatoTidspunktEksamenstypeStedLokale
2014-06-16 Ask your course manager Mundtlig eksamen ITU Preparation Room: 4A03
2014-06-16 Ask your course manager Mundtlig eksamen ITU 4A20
2014-06-17 Ask your course manager Mundtlig eksamen ITU 4A20
2014-06-17 Ask your course manager Mundtlig eksamen ITU Preparation Room: 4A03
2014-08-22 RE-EXAMIANTION - Ask your course manager Mundtlig eksamen ITU Preparation Room: 2A05
2014-08-22 RE-EXAMIANTION - Ask your course manager Mundtlig eksamen ITU 2A18