The student can:
- work with systems of linear equations, vectors, matrices, subspaces of the n-dimensional real space, and explain the connections between these concepts,
- work with determinants, eigenvalues, eigenvectors, linear transformations and connect them with the previous concepts.
In Linear Algebra we mainly focus on systems of linear equations (so-called linear systems). Many real-life situations can be modelled 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, and GPS.|
Linear Algebra starts with an introduction of linear systems which will be described using a (coefficient) matrix. You learn how to solve linear systems systematically, using a row reduction technique on the coefficient matrix.
Thereafter, we focus on operations for vectors and matrices, such as addition, multiplication, inverse and transpose. These operations are fundamental in Linear Algebra.
Next, we deal with sets of vectors with very nice properties: subspaces. It turns out that the properties of subspaces 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.
We also introduce the concept of determinant of a square matrix. We explore its properties and show some interesting interpretations.
Further, we deal with eigenvectors and eigenvalues of a square matrix. These concepts play a crucial role in discrete dynamic systems, which arise in many scientific fields.
Finally, we examine linear transformations and their properties. Some well-known applications in geometry will be treated as well.
Throughout, much emphasis is laid on the relations among the various concepts.