URL study guide
https://tue.osiris-student.nl/onderwijscatalogus/extern/cursus?cursuscode=2MBD50&collegejaar=2025&taal=enOmschrijving
This course is intended for students who are taking a double bachelor in mathematics and physics. It covers the content from the courses 2MBA20 and 2MBA50, Linear Algebra 1 and Linear Algebra 2.Vector spaces are a convenient way of modelling and working with space (and to generalize). Linear maps between vector spaces are functions that are ‘compatible’ with the linear structure of vector spaces. These concepts arise everywhere in mathematics, and we introduce them in an abstract setting so that the corresponding theory can be applied and used in various settings, for instance in other (mathematics) courses ranging from vector analysis to probability theory to optimization to machine learning. Linear algebra is also the starting point for studying more advanced non-linear objects and more advanced computational techniques (in other courses).
We will describe properties of vector spaces and inner product spaces (the latter being a vector space with some extra structure allowing us to speak of ‘distance’ and ‘angle’ between vectors), as well as properties of linear maps. We will see that linear maps can be given a matrix representation and see how using matrices allows us to solve systems of linear equations in a systematic way. We will analyze the nature of linear maps and learn how to find characteristic properties (called eigenvalues and eigenvectors). We will use these techniques to study interesting special classes of linear maps like orthogonal maps (linear maps that ‘preserve’ shapes, like reflections and rotations) and symmetric maps (which can be used to analyze quadratic curves and surfaces like ellipses and hyperboloids). The techniques covered in this course also enable us to decompose matrices into products of ‘simpler’ matrices in order to simplify certain mathematical tasks. E.g., such decompositions can be used to simplify the process of solving systems of linear equations like, e.g., certain types of linear differential equations.
Doelstellingen
ObjectivesThe student knows and understands the definitions of and (elementary) results about the notions mentioned in the following, and can
- reduce matrices to reduced row echelon form,
- solve systems of linear equations using row reduction,
- work with vectors in concrete and abstract vector spaces with and without inner products to determine linear (in)dependence,
- work with the concept of a linear map between vectors spaces, both for vector spaces that are handled as abstract objects and for concrete vector spaces,
- set up matrix representations of linear maps and work with both linear maps and matrix representations,
- deal with and perform basis transformations,
- carry out arithmetic operations with matrices, and determine the rank of a matrix,
- compute determinants,
- calculate eigenvalues, eigenvectors, eigenspaces and invariant subspaces for both linear maps and square matrices,
- work with the Jordan Normal Form of a matrix,
- work with important special classes of linear maps like symmetric and orthogonal ones,
- use eigenvalue/eigenvector techniques to analyse linear maps and quadratic forms in order to determine their (geometric) meaning,
- solve systems of linear differential equations using eigenvalue/eigenvector techniques,
- compute (orthonormal) bases
- solve problems in linear algebra by using and combining the various computational techniques
- develop proofs in situations regarding the various themes
- relate the various concepts and techniques in order to solve problems
The student is familiar with the document preparation system LaTeX and is able to write a short mathematical document in LaTeX according to (basic) scientific requirements.