Math105
Linear Algebra 1
Faculty
Matteo Levi
PostDoc , Universitat Autónoma de Barcelona and Universitat de Barcelona.
Course length
Duration
Total hours
Credits
Language
Course type
Fee for single course
Fee for degree students
Skills you’ll learn
Overview
This course provides a comprehensive introduction to linear algebra, a fundamental field of mathematics that encompasses vector spaces and linear transformations. Linear algebra is not just a key component of pure mathematics but also a potent tool with broad applications in computer science, physics, engineering, and data analysis. Proficiency in these concepts will furnish students with the capability to depict and tackle intricate issues, establishing the groundwork for subsequent studies and tangible implementations in machine learning, optimization, and scientific computation. Upon completion of this course, students will acquire both theoretical comprehension and hands-on problem-solving skills, readying them for advanced academic pursuits or practical real-world dilemmas.
Learning highlights
- This course aims to provide students with a solid understanding of the fundamental concepts of linear algebra and some of their practical applications.
- They will learn how to manipulate vectors and matrices and perform operations on them.
- More generally, they will learn how to use the language of linear algebra to formulate problems (such as systems of linear equations, for example) in matrix form. This is often the most convenient approach, as it is concise and computationally advantageous, as students will discover in coding courses.
- The focus will always be on the geometry of the problems: multiplying a matrix by a vector results in a geometric transformation in the vector's space, and keeping this in mind will enhance the understanding of the underlying processes when performing linear algebra operations.
Course outline
15 classes
Session 1
Vectors: linear combinations, linear independence, span, and basis.
Session 2
Vectors: dot product, lengths, angle between vectors, Schwarz, and triangle inequalities.
Session 3
Matrices: Matrix-vector multiplication, matrix-matrix multiplication, invertible matrices.
Session 4
Vector spaces and subspaces: column and row spaces, rank of a matrix. Understand matrices as linear transformations. Change of basis.
Session 5
Weekly review.
Session 6
Systems of linear equations: Gaussian elimination, pivoting, back substitution.
Session 7
Systems of linear equations: A=LU factorization, more on inverse matrices and inverse of elimination matrices, permutations and transpose matrices.
Session 8
Systems of linear equations: the nullspace and A=CR factorization. Full solution of Ax=b.
Session 9
More on independence, basis, and dimension. The relation between the four fundamental subspaces.
Session 10
Weekly review.
Session 11
Orthogonal subspaces and orthogonal projections. The least squares problem.
Session 12
Orthonormal bases, Gram-Schmidt process, A=QR factorization.
Session 13
Determinants, cofactors and inverse matrix computation.
Session 14
Exam.
Session 15
Exam.
Prerequisites
There are no specific prerequisites for attending the course, but any basic knowledge of algebra, trigonometry, and calculus would certainly help students follow the course more effectively.
Methodology
In each session, new concepts will be introduced. The theory will be complemented by some examples provided by the teacher and followed by an exercise session in which students will attempt to solve some simple exercises on their own, related to the concepts just introduced. The teacher will be available for any clarifications and may occasionally ask if a volunteer student would like to try presenting the solution to the exercise on the board for their peers. Students are expected to pay attention during the lectures and to actively work during the exercise session. Their dedication will influence the final grade of the course.
At the end of each session, some homework sheets will be assigned for students to practice at home. At the end of each week students are required to hand-in their solutions to the homeworks, which will be evaluated.
At the end of the course, we will have an oral exam. Students will be asked some questions about the course topics and will be required to solve step by step a few exercises selected from the homework sheets.
Grading
Matteo graduated in Mathematics from the University of Milan and subsequently obtained a Ph.D. from both the University of Bologna and the Universitat Autònoma de Barcelona, under a cotutelle agreement between the two universities.
In recent years, he has held research fellowships at Johns Hopkins University in Baltimore (U.S.), the Polytechnic University of Turin, and the University of Genoa. He currently holds a Margarita Salas PostDoc contract at the Universitat Autònoma de Barcelona.
See full profileApply for this course
Linear Algebra 1
by Matteo Levi
Total hours
45 Hours
Dates
Dec 02 - Dec 20, 2024
Fee for single course
€1500
Fee for degree students
€750
How to secure your spot
Complete the form below to kickstart your application
Schedule your Harbour.Space interview
If successful, get ready to join us on campus
FAQ
Will I receive a certificate after completion?
Yes. Upon completion of the course, you will receive a certificate signed by the director of the program your course belonged to.
Do I need a visa?
This depends on your case. Please check with the Spanish or Thai consulate in your country of residence about visa requirements. We will do our part to provide you with the necessary documents, such as the Certificate of Enrollment.
Can I get a discount?
Yes. The easiest way to enroll in a course at a discounted price is to register for multiple courses. Registering for multiple courses will reduce the cost per individual course. Please ask the Admissions Office for more information about the other kinds of discounts we offer and what you can do to receive one.