Mathematics: Algebra and Vector Geometry

<p>The overarching goal of the course is that the students acquire a deeper understanding of the foundations of algebra as well as knowledge of vector geometry and the foundations of linear algebra. Particular emphasis is put on developing the mathematical theory in a systematic manner contributing to the further aims of enhancing the students' ability to assimilate mathematical text, to carry out mathematical reasoning, to solve problems of both theoretical and applied character, and to communicate mathematics.</p><p>The course treats:</p> <ul> <li>Elementary logic and set theory</li> <li>Basic properties of the natural numbers and the integers: induction, divisibility, Euclid’s algorithm</li> <li>Basic properties of polynomials: divisibility, Euclid’s algorithm</li> <li>Complex numbers</li> <li>Linear systems of equations, Gaussian elimination</li> <li>Vectors in two and three dimensions, bases and coordinates, linear dependence, equations of lines and planes</li> <li>Scalar product, calculation of distances and angles</li> <li>Vector and volume product, calculation of area and volume</li> <li>Matrices, determinants, matrix invertibility.</li> </ul> <p>In addition, material on the axiomatic structure of the real numbers as well as the binomial theorem is treated during lectures taught jointly with the course Analysis in One Variable.</p>