Description:

Students will broaden their knowledge gained in the BIE-LA1 introductory course, where only vectors in the form of n-tuples of numbers were considered. Here we will introduce vector spaces in a general abstract form. The notions of a scalar product and a linear map will enable to demonstrate the profound link between linear algebra, geometry, and computer graphics. The other main topic will be numerical linear algebra, in particular problems with solving systems of linear equations on computers. The issues of numerical linear algebra will be demonstrated mainly on the matrix factorization problem. Selected applications of linear algebra in various fields will be presented.

Contents:

1. Abstract vector spaces, infinite-dimensional vector spaces.
2. Scalar products, vector norm, orthogonality.
3. Scalar products and analytical geometry.
4. [2] Linear maps and their matrices.
6. Affine transformations, homogeneous coordinates, projections and operations in 3D space as linear maps.
7. Introduction to numerical mathematics.
8. Solving systems of linear equations on computers.
9. [2] Matrix factorizations (LU, SVD, QR): computation and applications.
11. [3] Applications of linear algebra: the least-squares method, linear programming, recurrent equations.

Seminar contents:

1. Abstract vector spaces.
2. Scalar products, vector norm, orthogonality.
3. Analytical geometry.
4. Linear maps.
5. Matrices of linear maps.
6. [2] Affine transformations, homogeneous coordinates, projections and operations in 3D space as linear maps.
8. Systems of linear equations.
9. [2] Matrix factorizations (LU, SVD, QR).
11. The least-squares method.
12. Linear programming.
13. Recurrent equations.

Recommended literature:

1. Lloyd N. T., David B. : Numerical Linear Algebra. SIAM, 1997. ISBN 978-0898713619.
2. Lyche T. : Numerical Linear Algebra and Matrix Factorizations. Springer, 2020. ISBN 978-3030364670.
3. Gentle J. E. : Matrix Algebra: Theory, Computations and Applications in Statistics (2nd Edition). Springer, 2017. ISBN 978-3319648668.
4. Lengyel E. : Mathematics for 3D Game Programming and Computer Graphics (3rd Edition). Cengage Learning PTR, 2011. ISBN 978-1435458864.

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