Linear Algebra

Full course description

Linear algebra is the branch of mathematics which is primarily concerned with problems involving linearity of one kind or another. This is reflected by the three main themes around which this introductory course is centered.

               The first theme concerns what can be recognized without doubt as the most frequently occurring mathematical problem in practical applications: how to solve a system of linear equations. For this problem a complete solution procedure is developed which provides the student with a way to deal with such problems systematically, regardless of the number of equations or the number of unknowns.

               The second theme addresses linear functions and mappings, which can be studied naturally from a geometric point of view. This involves geometric ‘primitives’ such as points, lines and planes, and geometric ‘actions’ such as rotation, reflection, projection and translation.

One of the main tools of linear algebra is offered by matrices and vectors, for which a basic theory of matrix-vector computation is developed. This allows one to bring these two themes together in a common, exceptionally fruitful, framework. By introducing the notions of vector spaces, inner products, and orthogonality, a deeper understanding of the scope of these techniques is developed, opening up a large array of rather diverse application areas.

The third theme arises when the point of view is shifted once more, now from the geometric point of view to the dynamic perspective, where the focus is on the effects of iteration (i.e., the repeated application of a linear mapping). This involves a basic theory of eigenvalues and eigenvectors, which has many applications in various branches of science as will be discussed. For instance, important applications can be found in problems involving dynamics and stability, and applications to optimization problems found in operations research.

Many examples and exercises shall be provided to clarify the issues and to develop practical computational skills. They also serve to demonstrate practical applications where the results of this course can be successfully employed.

Students will obtain the insight that various seemingly different questions can all boil down to the same mathematical problem of solving a system of equations. Students will learn to look at the same problem from different angles and will learn to switch their point of view (from geometric to algebraic and vice versa).