Linear Algebra and Geometry I

5 credits

Syllabus, Bachelor's level, 1MA025

A revised version of the syllabus is available.

Code: 1MA025
Education cycle: First cycle
Main field(s) of study and in-depth level: Mathematics G1F
Grading system: Fail (U), Pass (3), Pass with credit (4), Pass with distinction (5)
Finalised by: The Faculty Board of Science and Technology, 19 March 2007
Responsible department: Department of Mathematics

Entry requirements

Basic Course in Mathematics

Learning outcomes

In order to pass the course (grade 3) the student should

know how to solve systems of linear equations using Gaussian elimination and be able to explain the relation between solvability and rank;

have a good command of matrix algebra, in particular know how to compute the inverse of a matrix, and know how to compute determinants of order 2 and 3;

be able to explain the basic properties of two- and three-dimensional vectors, master elementary vector algebra, be able to decide if vectors are linearly independent, and be familiar with the concepts of basis and coordinates;

be able to give an account of the concepts of scalar product and vector product, know how to compute such products and how to interpret them geometrically;

know the equations for a line and a plane and how to use these for computing intersections and distances;

be familiar with rotations, reflexions and orthogonal projections in two and three dimensions and be able to compute their matrices;

be able to interpret m×n matrices as linear transformations from Rⁿ to R^m;

be able to formulate important results and theorems covered by the course;

be able to use the theory, methods and techniques of the course to solve mathematical problems.

Content

Linear systems of equations: Gaussian elimination, rank, solvability. Matrices: matrix algebra and matrix inverse. Determinants of order two and three. Vector algebra, linear dependence and independence, bases, coordinates, scalar product and vector product, equations for lines and planes, distance, area and volume. Description of rotations, reflections and orthogonal projections in R² and R³. The linear space Rⁿ and m×n matrices as linear transformations from Rⁿ to R^m. The standard scalar product on Rⁿ and the Cauchy-Schwarz inequality.

Instruction

Lectures and problem solving sessions.

Assessment

Written examination at the end of the course. Moreover, compulsory assignments may be given during the course.