Description:

We will introduce students to the basic concepts of linear algebra, such as vectors, matrices, vector spaces. We will define vector spaces over the field of real and complex numbers and also over finite fields. We will present the concepts of basis and dimension and learn to solve systems of linear equations using the Gaussian elimination method (GEM) and show the connection with linear manifolds. We define the regularity of matrices and learn to find their inversions using GEM. We will also learn to find eigenvalues and eigenvectors of a matrix. We will also demonstrate some applications of these concepts in computer science.

Contents:

1. Fields, vectors, and vector spaces.
2. Matrices, matrix operations and matrix notation of a system of linear equations.
3. Solving systems of linear equations using Gauss elimination method.
4. Linear (in)dependence of vectors, linear span, subspace.
5. Base, dimension of a vector (sub)space.
6. Matrix rank, regularity of a matrix, inverse of matrix and its computation.
7. Frobenius theorem on solutions of a system of linear equations.
9. Linear manifolds, parametric and non-parametric equations of linear manifolds.
10. Permutations, determinant of a matrix.
11. [2] Eigenvalues and eigenvectors of matrices.
13. Diagonalization of matrices.

Seminar contents:

1. Matrices, matrix operations. Solving systems of linear equations using Gauss elimination method.
2. Linear (in)dependence of vectors, linear span, subspace. Base, dimension of a vector (sub)space.
3. Matrix rank, regularity of a matrix, inverse of matrix and its computation.
4. Frobenius theorem on solutions of a system of linear equations.
5. Linear manifolds, parametric and non-parametric equations of linear manifolds. Determinant of a matrix.
6. Eigenvalues and eigenvectors of matrices. Diagonalization of matrices.

Recommended literature:

1 Strang G. : Introduction to Linear Algebra (5th Edition). Wellesley-Cambridge Press, 2016. ISBN 978-0980232776.
2. Lay D.C., Lay S.R., McDonald J.J. : Linear Algebra and Its Applications (5th Edition). Pearson, 2015. ISBN 978-0321982384.
3. Axler S. : Linear Algebra Done Right (3rd Edition). Springer, 2014. ISBN 978-3319110790.
4. Klein P. N. : Coding the Matrix: Linear Algebra through Applications to Computer Science. Newtonian Press, 2013. ISBN 978-0615880990.

Keywords:

linear spaces, matrices, vectors, systems of linear equations

Abbreviations used:

Semester:

• W ... winter semester (usually October - February)
• S ... spring semester (usually March - June)
• W,S ... both semesters

Mode of completion of the course:

• A ... Assessment (no grade is given to this course but credits are awarded. You will receive only P (Passed) of F (Failed) and number of credits)
• GA ... Graded Assessment (a grade is awarded for this course)
• EX ... Examination (a grade is awarded for this course)
• A, EX ... Examination (the award of Assessment is a precondition for taking the Examination in the given subject, a grade is awarded for this course)

Weekly load (hours per week):

• P ... lecture
• C ... seminar
• L ... laboratory
• R ... proseminar
• S ... seminar