General Information

Course: Algebra 1
Language: English
ECTS credits: 7
Number of hours: 84
Main Teacher: Lucas GNECCO HEREDIA
Contact information: lucas.gneccoheredia@psl.eu
Tutorial teachers: Alice LACAN & Changhong WANG

Meeting times

6 hours per week, 2 days a week. The course is organized in the following blocks:

Lecture: Monday, 13h45 - 17h00
Tutorials: Friday, 09h00 - 12h15

Syllabus (course description)

This course is an introduction to linear algebra, with emphasis on the conceptual structures that underlie systems of linear equations, vector spaces, matrix methods, and linear transformations. Students will learn how to compute with matrices and vectors, how to reason abstractly about linear structures, and how to write full, rigorous mathematical solutions. The course balances computational fluency with theoretical understanding.

The course begins with systems of linear equations, matrices, and Gaussian elimination, then moves to determinants, vector spaces, basis and dimension, linear transformations and orthogonality. We then study eigenvalues, eigenvectors, and diagonalization. Throughout the semester, students will engage with both problem-solving and proof-based mathematics.

Learning Objectives

Mathematical Reasoning & Proofs

Interpret mathematical statements by identifying hypotheses and conclusions.
Determine whether a given example satisfies the assumptions of a definition or theorem.
Construct clear and rigorous proofs using direct arguments, contradiction, contrapositive reasoning, and induction when appropriate.
Write complete, logically ordered solutions to linear algebra exercises.

Linear Systems & Matrices

Solve systems of linear equations using Gaussian elimination and matrix methods.
Understand the relationship between system solvability, rank, and the existence/uniqueness of solutions.
Compute and interpret matrix operations, including addition, multiplication, transposition, and inversion.
Use determinants to characterize invertibility and understand geometric meaning such as scaling of volume.

Vector Spaces & Linear Transformations

Work with vectors, subspaces, span, linear independence, bases, and dimension.
Classify subsets as vector subspaces or not, with proofs or counterexamples.
Understand linear transformations and their matrix representations.
Compute kernels, images, and use the Rank–Nullity Theorem.
Analyze eigenvalues and eigenvectors and diagonalize matrices when possible.

Inner Products, Orthogonality & Applications

Work with dot products, norms, and inner products in finite-dimensional spaces.
Understand orthogonality, orthogonal complements, and projections onto subspaces.

Abstract & Applied Connections

Relate abstract results to concrete computations.
Recognize the geometric interpretation of linear algebra concepts.

Books and references

The course follows the textbook:

[Lang] Serge Lang. Introduction to Linear Algebra. Springer, 2nd edition.

The book is available on the PSL Explore catalogue, via Université PSL at:

http://catalogue.explore.psl.eu/permalink/f/125qeu0/TN_cdi_askewsholts_vlebooks_9781461210702

Additional useful references:

Gilbert Strang. Introduction to Linear Algebra.
Richard Penney. Linear Algebra: Ideas and Applications.
MIT OpenCourseWare: 18.06 Linear Algebra by Gilbert Strang.
3Blue1Brown, Essence of Linear Algebra (YouTube series).

Students are strongly encouraged to prepare the assigned reading before class in order to ask questions, identify difficult points, and engage actively with the material.

Evaluation

There will be one final exam and regular quizzes during the semester. The final grade will be computed as follows:

Evaluation	Weight
Final exam	80%
Quizzes	20%

All quizzes will have equal weight, and will be announced in advance via Moodle or via email.

Calendar

This calendar is a guideline and may be adjusted according to the pace of the class. Notice that each row corresponds to a block of 1.5h, meaning that the whole session of class in one day covers two rows. For example, the first session Monday 14-09-2026 is described in the first two rows.

Date	Session	Topic	Reading
Mon, 14-09-26	1.1	Course introduction and motivation (30 min). Vectors and points, dot product and norm, angle between vectors	Lang, Chapter 1, §1 - §4
Mon, 14-09-26	1.2	Parametric lines and planes	Lang, Chapter 1, §5 - §6
Mon, 21-09-26	2.1	Matrices, basic properties, product of matrices, matrix-vector product, matrix representation of linear systems, inverse matrix	Lang, Chapter 2, §1 - §2
Mon, 21-09-26	2.2	Homogeneous linear systems, elimination, row operations and Gaussian elimination	Lang, Chapter 2, §3 - §4
Mon, 28-09-26	3.1	Row operations and elementary matrices	Lang, Chapter 2, §5 - §6
Mon, 28-09-26	3.2	Vector spaces: basic definitions. Linear combinations. Convex sets	Lang, Chapter 3, §1 - §3
Mon, 05-10-26	4.1	Mock Exam # 1, Chapters 1-2
Mon, 05-10-26	4.2	Linear independence, basis and dimension	Lang, Chapter 3, §4 - §5
Mon, 12-10-26	5.1	Rank of a matrix	Lang, Chapter 3, §6
Mon, 12-10-26	5.2	Mappings, linear mappings	Lang, Chapter 4, §1 - §2
Mon, 19-10-26	6.1	Kernel and image of a linear map. Rank-nullity theorem. Rank and linear systems, solutions to linear systems, orthogonal complement	Lang, Chapter 4, §3 - §4
Mon, 19-10-26	6.2	Continuation of last course
Mon, 02-11-26	7.1	Matrix of a linear map	Lang, Chapter 4, §5 (with appendix)
Mon, 02-11-26	7.2	Composition of linear maps, inverses	Lang, Chapter 5, §1 - §2
Mon, 09-11-26	8.1	Mock Exam # 2, Chapters 3-5
Mon, 09-11-26	8.2	Scalar products, orthogonality, orthogonal bases	Lang, Chapter 6, §1 - §2
Mon, 16-11-26	9.1	Continuation of last course
Mon, 16-11-26	9.2	Determinants and its properties	Lang, Chapter 7, §1 - §2
Mon, 23-11-26	10.1	Determinants and linear dependence, Cramer’s rule	Lang, Chapter 7, §3 - §4
Mon, 23-11-26	10.2	Determinants and inverses, determinants as area and volume	Lang, Chapter 7, §5 - §6
Mon, 30-11-26	11.1	Mock Exam # 3, Chapters 6-7
Mon, 30-11-26	11.2	Eigenvectors and eigenvalues, characteristic polynomial	Lang, Chapter 8, §1 - §2
Fri, 11-12-26	12.1	Continuation of last course
Fri, 11-12-26	12.2	Symmetric matrices	Lang, Chapter 8, §3
Mon, 14-12-26	13.1	Diagonalization of a symmetric linear map	Lang, Chapter 8, §4
Mon, 14-12-26	13.2	Continuation of last course
Mon, 11-01-27	14.1	Mock Exam # 4, Chapter 8
Mon, 11-01-27	14.2	Review session: course roadmap