Aims

Module 1 Principles of Linear Algebra: Vector spaces

Learning goals - Students will be able to:

1. understand the concept of vector space
2. determine that a given transformation is a linear transformation or not
3. calculate all solutions of a system of linear equations and represent these in parametric form
4. determine the fundamental spaces of a matrix
5. verify linear (in)dependence, construct a basis for a given vector space
6. understand the concept of linear transformation, and be able to use the concepts of basis and coordinate system to represent a linear transformation as a matrix transformation
7. relate the fundamental spaces (incl properties such as dimension) of a matrix representing a transformation to the original (abstract) vector space

Module 2 Principles of Linear Algebra: Inner Product Spaces

Learning goals - The student will be able to:

1. compute the inner product and the norm and apply their properties
2. apply the Cauchy–Schwarz inequality, the triangular inequality and the Parallelogram Equality
3. use orthonormal bases and convert a vector from one base to another
4. apply the Gram-Schmidt procedure
5. define an Orthogonal Complement and an orthogonal projection
6. solve a minimization problem

Module 3 Linear Algebra: Eigenvalues

Learning goals - The student will be able to:

1. define and calculate eigenvectors, eigenvalues and their algebraic and geometric multiplicities
2. understand and exploit the connection between eigenvalues, the determinant and the matrix trace
3. calculate an eigenvalue decomposition
4. understand Hermitian matrices admit an orthogonal eigendecomposition
5. calculate the singular value decomposition and low rank approximations to linear transformations

Module 4 Integral Theorems

Learning goals - The student will be able to:

1. understand and use the concept of vector fields and conservative vector fields
2. calculate line integrals of vector fields (and to simplify such integrals by singling out the conservative parts, if any
3. calculate surface integrals and flux integrals
4. understand the physical meaning of gradient, divergence, curl operators
5. calculate the gradient of a scalar function and the divergence and curl operators applied to vector functions
6. understand and apply vector identities/calculus
7. understand and apply the theorem of Green

Module 5 Optimization in multiple variables

Learning goals - The student will be able to:

1. understanding of the concept of a real function of n real variables
2. understanding of the concept of partial derivatives and ability to compute them for a given function
3. understanding of the concept of gradient and its link to differentiability of a function of n real derivatives
4. understanding of the concepts of the chain rule and directional derivative
5. ability to apply the chain rule and compute the directional derivative of a given function of n real derivatives
6. understanding of the relation between critical points and extrema of a function of n real derivatives
7. determine the extrema of a function of n real derivatives
8. understanding of the concept of optimization with (a) side condition(s) and ability to solve such given problems
9. understanding of the concept of Lagrange multipliers and ability to exploit them to determine the solution of optimization problems with one or more side conditions

Module 6 Differential Equations

Learning goals - The student will be able to:

1. classify differential equations and systems of differential equations based on properties such as order, dimension, variable coefficients and linearity 3
2. solve first order linear differential equations
3. solve systems of linear differential equations with constant coefficients
4. understand and apply linear algebra techniques to linear systems of differential equations with constant coefficients
5. analyse the stability of linear systems and some non-linear systems

Is included in these courses of study

• Voorbereidingsprogramma: Master in de ingenieurswetenschappen: wiskundige ingenieurstechnieken (Leuven) 29 ects.
• Master in de ingenieurswetenschappen: werktuigkunde (programma voor studenten gestart vóór 2024-2025) (Leuven) 120 ects.
• Master of Mechanical Engineering (Leuven) 120 ects.
• Master of Materials Engineering (Leuven) 120 ects.