The student is able to:
- work with systems of linear equations, vectors, matrices,subspaces of Rn and explain the connections between these concepts
- work with determinants, eigenvalues, eigenvectors,linear transformations and connect them with the previous concepts
In this course, we mainly focus on systems of linear equations (linear systems). Many real-life situations can be modeled as a linear system. Examples are networks (traffic networks, data networks, electrical networks, etc.), economic models, chemical reactions, cryptography (coding of messages), scheduling, computer graphics, GPS.
Linear Algebra starts with an introduction of linear systems which will be described using a (coefficient-)matrix. Already in the first week we learn how to solve linear systems systematically, using a row reduction technique on the coefficient matrix.
In the second week we focus on operations vectors and matrices, such as addition, multiplication, inverse and transpose. These operations are a fundamental issue in Linear Algebra.
In the third week we deal with sets of vectors with very nice properties: subspaces. It turns out that the properties of subspaces can tell us a lot about the structure of solution sets of linear systems. Here the concepts of linear combination, linear independence, basis and dimension play an important role.
In the fourth week we introduce the concept of determinant of a square matrix. We explore its properties and show some interesting interpretations.
The fifth week will be about eigenvectors and eigenvalues of a square matrix. These concepts play a crucial role in discrete dynamical systems, which arise in many scientific fields.
Finally, in the last week, we examine linear transformations and their properties. Some well-known applications in geometry will be treated as well.
Much emphasis is laid on the relations among the various concepts.
A case study may be implemented in each program, to become acquainted with applications of Linear Algebra.