Learn fundamental mathematical techniques from Linear Algebra and Calculus used in STEM domains, critically reflect on these through pertinent examples, and practice the concepts with the use of applets and exercises.
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Mathematics is the most essential tool in any STEM professional’s toolbox. In this course, we will provide you with an introduction to linear algebra, multivariable calculus, and differential equations, through exploring the main definitions, theorems and practical examples required.
Can we use linear algebra to do data compression? What’s the meaning of an eigenvalue and an eigenvector in a mechanical system? How do vector fields help to describe wind flow? How can you make optimal parameter choices in industrial processes?
We aim to answer all these questions and more, so that you can use these mathematical techniques when tackling problems in your own field of study.
We will use examples, graphic representations, applets, and exercises to exemplify the various theorems and definitions.
You will acquire the skills to cope with matrix-formulated problems typically arising from applications in science and technology. Not only will you be able to use practical algorithms, solve systems of equations and differential equations, compute the singular value and eigenvalue decomposition, and solve optimisation problems, you will also acquire a set of properties that will assist in simplifying and understanding mathematical problems.
The course will give you the tools to transform optimisation problems and differential equations into matrix language. Most importantly, you will learn that matrix computations are ubiquitous in science and engineering.
What you'll learn
- What vector spaces are and how their elements can be represented by coordinate vectors with respect to a basis
- Linear transformations between vector spaces and how to represent them in matrix notation
- To compute inner products, norms, and orthogonal projections
- To define and calculate eigenvalues and eigenvectors and their algebraic and geometric multiplicities
- To calculate the singular value decomposition
- To understand the concepts of a real function of multiple variables, partial and directional derivatives and the multivariate chain rule
- To determine critical points and identify extrema of multivariate functions
- To understand the concepts of (conservative) vector fields and be able to calculate and simplify their line integrals
- To understand what gradient, divergence, and curl operators are and how to calculate them
- To classify and solve (systems of) first-order differential equations
-To understand and apply linear algebra techniques to solve linear systems of differential equations with constant coefficients and analyse their stability
Syllabus
- Vector Spaces
- Vector Spaces
- Basis and Coordinates
- Fundamental Spaces
- Linear Transformations
- Inner Product Spaces
- Inner Product and Norm
- Projection and Orthogonal Bases
- Least Squares
- The Eigenvalue Decomposition
- Eigenvalues and Eigenvectors
- Theorem and Properties
- The Eigenvalue Decomposition
- Properties of Symmetric Matrices
- The Singular Value Decomposition
- Optimisation
- Real Functions of n Real Variables
- Curves in Rn
- Partial Derivatives and Gradient
- Extrema
- Integral Theorems
- Vector and Scalar Fields
- Conservative Vector Fields
- Line Integrals of Vector Fields
- Double Integrals
- Gradient, Divergences, Curl Operators
- Theorem of Green
- Differential Equations
- First Order Differential Equations
- Linear Systems of First Order Differential Equations
- Non-Linear Autonomous Systems
