Foundation in Linear Algebra

Linear algebra provides many tools and techniques that are used to solve typical problems in Data Science. Wherever larger collections of numbers are involved, linear algebra can provide solutions. Linear algebra is used in regression problems, in deep learning, in recommender systems, in search engines (page rank), in unsupervised learning (PCA, SVD, NMF...), etc. Here, students acquire the basic skills of linear algebra and build a solid understanding of its tools (matrices and vectors) and its applications (systems of equations, vector geometry, linear mappings). They are able to perform all necessary calculations using modules of Python.

Linear systems of equations Students are able to write down systems of linear equations in matrix form and solve them using the Gaussian method. They can use the rank of a matrix to determine the number of solutions to a system of equations and interpret systems of equations with 2 and 3 variables geometrically.

Matrices Algebra Students understand the concept of a matrix and know the most important matrix operations and their properties and are familiar with the different special matrices. They understand how to calculate the multiplicative inverse of a matrix and use it to solve systems of linear equations. Students can determine the determinant of matrices using Sarrus' rule or Laplace's theorem, as appropriate, and understand the connection to solvability of systems of linear equations. They can use Cramer's rule to solve systems of linear equations.

Vector geometry Students understand the concept of a vector and know the common vector operations and their properties. They are familiar with the definition and properties of the scalar product and applications such as checking vectors for orthogonality, calculating angles between vectors and orthogonal projections. They know the definition and properties of the cross product and understand the relationship to calculating parallelogram areas. They can determine the spat product of three vectors and know its geometric interpretation.

Students can describe straight lines and planes (where possible) in a parametric, coordinate, normal and Hessian normal form and can convert these into each other. They can use these representations to determine the distance between points, straight lines and planes and calculate the intersection of two given objects.

Vector spaces and linear mappings Students know the properties of real Euclidean vector spaces and can extend these concepts to general real vector spaces. They understand the difference between a generating system and a basis and the concept of linear independence.

They understand the concept of mappings between vector spaces, in particular vector space homorphisms, which they can describe with matrices and able to invert and link together. They are familiar with the most common linear transformations (rotation, reflection, orthogonal projection, stretching/compression).

Python Students are be able to use the numpy, sympy, and matplotlib packages to implement and, if necessary, visualize the linear algebra concepts covered.