6 hours per week, 3 days a week. There are four sessions per week divided into the following blocks:
The amount of sessions dedicated to exercise sessions can change throughout the semester, but it should be around 2 blocks per week.
This course is a first introduction to linear algebra. Students will learn basic notions regarding vectors, matrices, systems of equations and linear transformations. The first part of the course is more centered on computations and manipulation of matrices, and as the courses advances, more focus is put on the theoretical aspects of vector spaces. An important emphasis will be made on writing mathematics properly in the way of proofs and complete solutions to exercises. Throughout the course, applications of linear algebra to data, signal processing and differential equations will be studied.
Mathematical Reasoning & Proofs
Linear Systems & Matrices
Vector Spaces & Linear Transformations
Abstract & Applied Connections
The structure of the course will follow closely the first reference. The letters at the beginning are used to identify the book when proposing readings and exercises on the calendar of the course.
Ideally, each session will cover a particular part of a reference that should be read/viewed by students before class. Students are strongly encouraged to do this in order to have questions and comments.
There are also online resources that can be very useful for students:
There will be one final exam on the dedicated exam week at the end of the semester. During the semester, there will be two more partial exams on dates to be defined (the dates in the table are tentative). There will also be weekly quizzes that will provide points, out of which a final grade will be computed. Evaluation will test students on the different learning objectives.
The final grade will be computed on the following way:
| Evaluation | Date | Weight |
|---|---|---|
| Partial exam 1 | 2025-10-20 (Monday) | 25% |
| Partial exam 2 | 2025-12-15 (Monday) | 25% |
| Quizzes | All semester | 10% |
| Final exam | Week starting 2026-02-02 | 40% |
This is a guideline for the course that can be adapted depending on the development of the course. It is based on a total of 60 sessions.
| Session | Topics | Readings |
|---|---|---|
| 2025-09-22 (Mon) - 1 | Matrices, row/column vectors; Linear combinations, span, dependence; Definition of vector space | [RP1] §1.1; [GS2] §1.1 |
| 2025-09-23 (Tue) - 2 | Lengths and dot products (intro) | [GS2] §1.2 |
| 2025-09-24 (Wed) - 3 | Exercise session | |
| 2025-09-24 (Wed) - 4 | Exercise session | |
| 2025-09-25 (Thu) - 5 | Linear equations, linear systems; Existence/uniqueness; Gaussian elimination; Rank of a matrix | [RP1] §1.2; [GS2] §2.1 |
| 2025-09-25 (Thu) - 6 | Exercise session | |
| 2025-09-29 (Mon) - 7 | Gaussian elimination; Row operations and row equivalence; Echelon and RREF | [RP1] §1.3; [GS2] §§2.2–2.3 |
| 2025-09-30 (Tue) - 8 | Application to circuits and traffic flow | [RP1] §1.2.2, §1.3.2 |
| 2025-10-01 (Wed) - 9 | Exercise session | |
| 2025-10-01 (Wed) - 10 | Exercise session | |
| 2025-10-02 (Thu) - 11 | Subspaces; Column space and null space | [RP1] §1.4; [GS2] §§3.1–3.3 |
| 2025-10-02 (Thu) - 12 | Exercise session | |
| 2025-10-06 (Mon) - 13 | Test for linear independence; Basis and dimension of subspaces | [RP1] §§2.1–2.2; [GS2] §3.4 |
| 2025-10-07 (Tue) - 14 | Applications to differential equations | [RP1] §§2.2.2 |
| 2025-10-08 (Wed) - 15 | Exercise session | |
| 2025-10-08 (Wed) - 16 | Exercise session | |
| 2025-10-13 (Mon) - 17 | Row space and rank; Rank-nullity theorem; Singular matrices and solutions | [RP1] §2.3; [GS2] §3.5 |
| 2025-10-14 (Tue) - 18 | Continue last session | |
| 2025-10-15 (Wed) - 19 | Exercise session | |
| 2025-10-15 (Wed) - 20 | Exercise session | |
| 2025-10-20 (Mon) - 21 | Partial Exam 1 | - |
| 2025-10-21 (Tue) - 22 | Linear transformations; Matrix of a linear transformation; Matrix multiplication as composition | [RP1] §§3.1–3.2; [GS2] §§8.1–8.2 |
| 2025-10-22 (Wed) - 23 | Inverse of a transformation; Inverse matrix via elimination | [RP1] §3.3; [GS2] §2.5 |
| 2025-10-22 (Wed) - 24 | Exercise session | |
| 2025-11-03 (Mon) - 25 | LU factorization; Permutation matrices | [RP1] §3.4; [GS2] §§2.6–2.7 |
| 2025-11-04 (Tue) - 26 | Bases and change of basis; Isomorphisms; Invertible linear transformations; Dimensions of subspaces | [RP1] §3.5; [GS2] §§3.4–3.5 |
| 2025-11-05 (Wed) - 27 | Exercise session | |
| 2025-11-05 (Wed) - 28 | Exercise session | |
| 2025-11-10 (Mon) - 29 | Determinants and properties | [RP1] §4.1; [GS2] §§5.1–5.2 |
| 2025-11-12 (Wed) - 30 | Determinant results; Cramer’s rule | [RP1] §§4.2–4.3; [GS2] §5.3 |
| 2025-11-12 (Wed) - 31 | Exercise session | |
| 2025-11-17 (Mon) - 32 | Eigenvectors and eigenvalues | [RP1] §5.1; [GS2] §6.1 |
| 2025-11-18 (Tue) - 33 | Application to Markov Processes | [RP1] §5.1.2 |
| 2025-11-19 (Wed) - 34 | Exercise session | |
| 2025-11-19 (Wed) - 35 | Exercise session | |
| 2025-11-25 (Tue) - 36 | Diagonalization | [RP1] §5.2; [GS2] §6.2 |
| 2025-12-01 (Mon) - 37 | Applications to systems of differential equations | [RP1] §5.2.2 |
| 2025-12-02 (Tue) - 38 | Orthogonality; Orthogonal/orthonormal bases | [RP1] §6.1; [GS2] §4.1 |
| 2025-12-03 (Wed) - 39 | Exercise session | |
| 2025-12-03 (Wed) - 40 | Exercise session | |
| 2025-12-08 (Mon) - 41 | Projections; Orthogonal complement | [RP1] §6.2; [GS2] §4.2 |
| 2025-12-09 (Tue) - 42 | Gram-Schmidt process; QR decomposition | [RP1] §6.2; [GS2] §4.4 |
| 2025-12-10 (Wed) - 43 | Exercise session | |
| 2025-12-10 (Wed) - 44 | Exercise session | |
| 2025-12-15 (Mon) - 45 | Partial Exam 2 | |
| 2025-12-16 (Tue) - 46 | Scalar product spaces; Fourier series | [RP1] §6.3; [GS2] §10.5 |
| 2025-12-17 (Wed) - 47 | Application: data compression (wavelets) | [RP1] §6.3.1 |
| 2025-12-17 (Wed) - 48 | Exercise session | |
| 2026-01-05 (Mon) - 49 | Orthogonal matrices | [RP1] §6.4 |
| 2026-01-06 (Tue) - 50 | Least squares | [RP1] §6.5; [GS2] §4.3 |
| 2026-01-07 (Wed) - 51 | Exercise session | |
| 2026-01-07 (Wed) - 52 | Exercise session | |
| 2026-01-12 (Mon) - 53 | Quadratic forms; Symmetric matrices; Spectral Theorem; Positive definite matrices | [RP1] §6.6; [GS2] §§6.4–6.5 |
| 2026-01-13 (Tue) - 54 | Singular Value Decomposition | [RP1] §6.7; [GS2] §§7.1–7.2 |
| 2026-01-14 (Wed) - 55 | Exercise session | |
| 2026-01-14 (Wed) - 56 | Exercise session | |
| 2026-01-19 (Mon) - 57 | Geometry of the SVD | [GS2] §7.4 |
| 2026-01-20 (Tue) - 58 | Generalized eigenvectors; Jordan form | [RP1] §§7.1–7.2; [GS2] §8.3 |
| 2026-01-21 (Wed) - 59 | Exercise session | |
| 2026-01-21 (Wed) - 60 | Exercise session |