91157 - Linear Algebra

Learning outcomes

: By the end of the course, the successful student has the ability to perform basic operations with vectors and matrices, to compute determinants, to compute Eigen values and Eigen vectors and to solve linear systems.

Course contents

Linear algebra is one of the fundamental languages of modern science. It provides systematic methods for representing, transforming and interpreting complex data, and underlies much of numerical modelling, statistics, data analysis and machine learning. In genomics, datasets such as gene-expression profiles, sequence-derived features and population data are naturally organised in terms of vectors and matrices.

 

The course begins with systems of linear equations and their matrix representation. Gaussian elimination, row-echelon forms, the rank of a matrix and the rank criterion for the existence and uniqueness of solutions will be discussed.

Basic matrix algebra will then be introduced, including matrix addition and multiplication, transposition, invertibility and the relation between invertible matrices and uniquely solvable linear systems.

The course will next introduce finite-dimensional vector spaces and subspaces. The main topics will be linear combinations, spanning sets, linear independence, bases, coordinates and dimension.

Linear transformations will be studied together with their matrix representations. Particular attention will be given to kernels, images, rank, the rank-nullity theorem, composition and invertibility.

Determinants will be introduced, together with their principal properties and computational methods, including cofactor expansion. Their relation to invertibility and linear transformations will also be discussed.

The final part of the course will cover changes of basis, similar matrices, eigenvalues, eigenvectors, eigenspaces and diagonalizability.

Throughout the course, selected examples will illustrate how vectors, matrices and linear transformations arise in scientific applications and in the representation and analysis of data, with occasional references to problems relevant to genomics.

Inner products, orthogonality and the spectral theorem are not part of the course syllabus.

 

By the end of the course, students will be able to:

• solve systems of linear equations using Gaussian elimination and describe their sets of solutions;
• perform basic computations with matrices, including the computation of inverses, ranks and determinants;
• work with finite-dimensional vector spaces and subspaces, linear combinations, spanning sets, linear independence, bases and dimension;
• represent linear transformations by matrices and compute their kernels, images and ranks;
• perform changes of basis and understand the relation between different matrix representations of the same linear transformation;
• compute eigenvalues and eigenvectors, determine whether a matrix or linear transformation is diagonalizable, and carry out a diagonalization when possible;
• recognise and interpret elementary uses of vectors, matrices and linear transformations in scientific and data-oriented problems.
  

 

 

Readings/Bibliography

Main textbook

 R. Fioresi and M. Morigi, Linear Algebra, CEA.

Open-access companion textbook

 J. Hefferon, Linear Algebra. The electronic edition is freely available from the author's website.

Optional further reading

 R. Ghrist, Linear Algebra: Essence & Form. Freely available online. Selected chapters may be useful for additional examples and applications; several topics covered in this book are beyond the scope of the course.

 

Additional notes, exercises and teaching materials may be made available through the Virtuale platform.

Teaching methods

The course is taught in English.

Classes consist mainly of lectures, including worked examples and guided problem solving. The presentation will combine the essential theoretical foundations with computational methods and concrete examples. The level of abstraction will be kept appropriate for students whose main field of study is not mathematics.

Assessment methods

The assessment is based on a written examination and, in the cases specified below, an oral examination.

The written examination lasts three hours and consists of exercises and questions concerning both computational methods and the theoretical concepts discussed during the course.

Students who obtain a written mark between 18/30 and 25/30 are not required to take an oral examination and may accept the written mark as their final mark.

Students who obtain a written mark of 26/30 or higher are required to take an oral examination. The oral examination may include a discussion of the written paper and questions on any of the topics covered during the course.

Admission to the oral examination requires a passing mark in the written examination.

Teaching tools

Chalk and blackboard.

 

Lecture handouts may be provided.

Office hours

See the website of Alessandro D'Andrea

See the website of Stefano Riolo

See the website of Alessia Cattabriga