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Elements of Linear Algebra

Constructor University, Fall 2025

Official Class Description from Campusnet

This module is the first in a sequence introducing mathematical methods at the university level in a form relevant for study and research in the quantitative natural sciences, engineering, Computer Science. The emphasis in these modules is on training operational skills and recognizing mathematical structures in a problem context. Mathematical rigor is used where appropriate. However, a full axiomatic treatment of the subject is provided in the first-year modules "Analysis" and "Linear Algebra". The lecture comprises the following topics

• Review of elementary analytic geometry
• Vector spaces, linear independence, bases, coordinates
• Matrices and matrix algebra
• Solving linear systems by Gauss elimination, structure of general solution
• LU decomposition and matrix inverse
• Linear maps and connection to matrices
• Determinant
• Eigenvalues and eigenvector
• Hermitian and skew-Hermitian matrices
• Orthonormal bases, Gram-Schmidt orthonormalization and QR decomposition
• Fourier transform
• Singular value decomposition
• Principal Component Analysis and best low rank approximations

News

• The first tutorial is on Wed., Sep. 10, 2025 in East Wing (IRC). There is no tutorial on Sep 3.
• First class is on Mon., Sep. 1, 2025, 14:15, in RLH-172 (Conrad Naber Lecture Hall).

Contact Information

Time and Place

How is this class organized?

In each week, you are supposed to:

• As preparation or in case you cannot make it to class, watch the videos of the two corresponding sessions via MS Teams (links below). Note that the videos were recorded for the 2024 version of this class, so they might not contain some minor improvements from the lecture notes.
• Come to class on Mondays and Tuesdays at 14:15 (held by the instructor).
• Come to the tutorial on Wednesdays (held by the TAs).
• Work on the moodle exercises/quizzes (online multiple choice questions) and submit them before the following tutorial.
• Every other week, work on the homework sheets (linked below as pdfs), and submit your handwritten solution to moodle on the due date before the tutorial.

Textbooks

• Gilbert Strang: Introduction to Linear Algebra (Fifth Edition, 2016). (Available as e-book in the library.)
• S.A. Leduc: Linear Algebra (Hoboken: Wiley, 2003). (Available as e-book in the library.)
• Riley, Hobson, Bence (RHB): Mathematical Methods for Physics and Engineering (3rd edition, Cambridge, 2006). (Physical copies available in the library.)

Table of Contents

Chapter 1: Basic Calculus Review
1.1: Numbers and Polynomials
1.2: Functions

Chapter 2: Vectors and Vector Spaces
2.1: Elementary Analytical Geometry
2.2: Vector Spaces

Chapter 3: Matrices and Linear Equations
3.1: Matrices and Linear Maps
3.2: Systems of Linear Equations
3.3: Matrix Inverse

Chapter 4: Determinants

Chapter 5: Eigenvalues and Eigenvectors
5.1: Eigenvalues/vectors
5.2: Eigenspaces
5.3: Diagonalization

Chapter 6: Special Types of Matrices
6.1: Normal Matrices
6.2: Hermitian/Self-adjoint Matrices
6.3: Real Symmetric Matrices
6.4: Unitary and Orthogonal Matrices

Chapter 7: Matrix Decompositions
7.1: LU Decomposition
7.2: QR Decomposition
7.3: Singular Value Decomposition
7.4: Principal Component Analysis and Best Low-Rank Approximation

Grading

The grade is only based on the final exam. Moodle-quizzes and bi-weekly homework submissions can each provide up to 5% bonus points (i.e., up to 10% bonus points in total can be achieved) according to the following table:

HW percentage solved	Bonus percentage
80 or more	5
60 - 79	4
40 - 59	3
20 - 39	2
5 - 19	1
less than 5	0

Exams

There will be one final exam (centrally scheduled in December) and one make-up final exam (centrally scheduled in January).

Links to previous exams will be provided closer to the exam date.

Practice, Practice, Practice

An essential component for doing well in this class is to work on practice exercises. Math is about problem solving (as are almost all sciences)! During this course lots of possibilities for solving exercises are provided on moodle, in the example sessions, and in the tutorial, see below.

Moodle Exercises

Please go to moodle, login, and select the Elements of Linear Algebra class to view the exercises, and the solutions (after the due date). Each week on Monday a new quiz is released, and this is due the following week before the tutorial.

Homework Exercises

These are released bi-weekly, and scans of handwritten solutions are to be uploaded to moodle before the due date.

Extra Material

• Learn the Greek alphabet with this wikipedia article.
• Extra notes on set and interval notation from Week 1.

Class Schedule

Will be updated while class is progressing.

Below, please click on the date to download the lecture notes of this day.

(Note that the book references given below offer only a rough orientation. Sometimes, only parts of a particular chapter are covered in class.)

Note that the videos were recorded for the 2024 version of this class, so they might not contain some minor improvements from the lecture notes.

Date	Topics
Week 1 (Sep. 1 - 7, 2025)
Session 1 Notes Video (2024)	Topic: Review of natural, rational, real, and complex numbers You will learn about the following topics: Natural numbers, integers, rational numbers, real numbers, complex numbers Polynomials and their roots Irrationality of square root of 2 Fundamental Theorem of Algebra Literature: any Calculus textbook, RHB 1.1
Session 2 Notes Video (2024)	Topic: Functions, their inverses, and their graphs You will learn about the following topics: Definition of function, domain and range Discussion of standard functions: absolute value, parabola, hyperbola, sin, cos, tan, exponential function Inverse of a function Literature: any Calculus textbook
Extra Examples	More on set notation, Complex numbers, Roots of Polynomials, Roots of quadratic equations, Logarithm, Inverse Functions
Extra Material	Sumary of the notation I have used in Week 1 for sets and intervals.
pdf of moodle quiz	Please submit on moodle
pdf of homework sheet	Covering Weeks 1 and 2. Please submit on moodle
Week 2 (Sep. 8 - 14, 2025)
Session 3 Notes Video	Topic: Vectors in Euclidean space, vector operations, scalar product, cross product You will learn about the following topics: Vectors and their basic operations Length of a vector Unit vectors Scalar product and its geometrical interpretation Cross product and its geometrical interpretation Literature: selected topics from RHB Chapter 7
Session 4 Notes Video	Topic: Lines and planes You will learn about the following topics: Lines and their parametrizations Lines defined by linear equations Planes and their parametrizations Planes defined by normal vectors Planes defined by linear equations First encounter with systems of linear equations Literature: RHB 7.7
Example Session	Scalar and cross products, Vector application: centroid of a triangle, Lines and planes
pdf of moodle quiz	Please submit on moodle