Objectives and competences
Objectives:
Thoroughly learn, understand and acquire in-depth theoretical knowledge of the basics of linear algebra and their application in engineering mathematics.
Competences:
Ability to combine mathematical knowledge and skills in linear algebra problem solving.
The ability to use the acquired knowledge for computing some quantities in engineering and technics which appear in different basic and applied sciences, for example Mathematical analysis II, Physics, Electrical engineering, Thermodynamics.
Content (Syllabus outline)
1. Vectors in space: definition, geometrical interpretation, basic operations, scalar product, vector product, mixed product, analitical geometry in space – straightlines and planes, connection with physics.
2. Matrices: operations, different kinds of matrices, inverse matrix, determinant, systems of linear equations, Gauss elimination, Cramer's rule, matrix rank.
3. Vector spaces: spaces and subspaces, basis, dimension, examples.
4. Linear transformations: basic properties and examples, kernel and image, spaces of linear transformations, matrix as a linear transformation, matrix representation of a linear transformation, special cases of linear transformations - rotations, mirroring, projections.
5. Eigenvalues and eigenvectors: basic properties and examples, characteristic polynomial, diagonalization of a matrix.
Learning and teaching methods
Lectures, tutorial, coursework assignments.
Intended learning outcomes - knowledge and understanding
Knowledge and understanding:
At the end of the course students are going to:
1. Perform basic arithmetic operations with matrices and vectors (addition, subtraction, product of the vector by scalar, transposition, multiplication).
2. Describe scalar, vector, and mixed product of vectors and use them correctly in calculating the angle between vectors, vector length, figure area, body volume, checking the perpendicularity, parallelism, and linear dependence of vectors.
3. Explain the equations of basic objects (straightline, plane) of analytical geometry in the space, condider mutual positions between objects and calculate the distances between them.
4. Identify and describe different types of matrices.
5. Calculate the determinant of an arbitrary dimension matrix and list the properties of the determinant and illustrate them with examples.
6. Calculate the inverse of the matrix using an extended matrix or an adjunctive matrix and use it to solve matrix equations.
7. Solve systems of linear equations using Gaussian elimination and Cramer's rule and consider their solutions with respect to the given parameter in the system.
8. Be able to understand vector spaces and subspaces and describe their properties.
9. Define notions related to linear transformations and use them at matrix representation of a linear transformation. Use the basic transformations of a plane (rotation, mirroring, projection) and their composite to map a vector or point.
10. Define notions related to eigenvalues and eigenvectors of linear transformations and matrices and find the eigenvalues and the corresponding eigenvectors of the matrix and study the diagonalizability of the matrix.
Intended learning outcomes - transferable/key skills and other attributes
Transferable/key competences and other abilities:
• Communication skills: unequivocal and precise expression.
• Computational skills: performing a variety of computational operations.
• Problem solving: identifying and solving mathematical problems that occur in energy and that are described by the theory of linear algebra.
Readings
Osnovna/Basic:
- M. Mencinger, Zbirka rešenih nalog iz matematične analize in algebre, FG UM, Maribor, 2006.
- R. Jamnik, Matematika, DMFA Slovenije, Ljubljana, 2008.
Dodatna/Additional:
- I. Vidav, Višja Matematika I, DMFA Slovenije, Ljubljana, 2008.
- E. Kreyszig, Advanced Engineering Mathematics, J. Wiley and Sons, 2011.
Additional information on implementation and assessment Method (written or oral exam, coursework, project):
• Written exam
• Oral exam
• coursework
Notes:The student must finish each part of the exam (written examination, oral exam, courseworks with at least 50%. In the case that a student is offered the opportunity to take a written exam in the form of two midterm exams, each must be graded with at least 30% and both with an average of at least 50% to pass the written part of the exam.
