| Kursusnavn (dansk): | Mathematical Analysis and Linear Algebra |
| Kursusnavn (engelsk): | Mathematical Analysis and Linear Algebra |
| Semester: | Forår 2015 |
| 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
• Compute Taylor series
• Test an infinite series for convergence
• Compute derivatives of curves, compute arc lengths
• Compute gradients, directional derivatives and tangent surfaces etc
• Find minima and maxima of functions of many variables, also under constraints, using Lagrange multipliers
• Compute double integrals
• Solve systems of linear equations using elimination, and to find the complete solution to such a system
• Use the method of least squares
• Compute inverses of matrices
• Compute the dimensions of the four fundamental subspaces, and compute bases for these
• Compute projection matrices
• Compute orthonormal bases of finite dimensional vector spaces
• Compute determinants of small matrices and compute cofactor matrices
• Compute eigenvectors, eigenvalues and diagonalise matrices
Moreover, for all of the above, the student should be able to describe relevant theory and repeat relevant definitions. The student should be able to argue informally for the correctness of all the methods used. |
| 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.
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. For this reason the course has an oral exam. |
| Læringsaktiviteter: | 14 ugers undervisning bestående af forelæsninger og øvelser TBA |
| Obligatoriske aktivititer: | Content
Workload
Activities
There will be 6 mandatory out of which 5 must be approved.
Feedback
There will be written feedback
What if the student fails to pass a mandatory activity:
A resubmission will be possible
Be aware: The student will receive the grade -03 at the ordinary exam, if the mandatory activities are not approved and the student will use an exam attempt. |
| Eksamensform og -beskrivelse: | X. experimental examination form (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.
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| Litteratur udover forskningsartikler: | - Gilbert Strang: Introduction to Linear algebra, 4th edition
Calculus book to be decided. |
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Kursusoversigt