## Linear algebra over the integers

Goal:

Linear algebra is the study of matrices, linear maps, vector spaces, etc. Usually one chooses the coefficients to be in a field, such as the rational or the real numbers, finite fields, and others.

For many applications, for instance in number theory and cryptography, one wants to work with matrices over the integers. This means, in particular, that one must not divide (otherwise one usually ends up with a rational number with a non-trivial denominator)! Instead, the greatest common divisor plays an important role. If one adapts the Gauss algorithm accordingly, one can transform matrices into the Hermite Normal Form, which can be used for solving systems of linear equations.

The project allows many different subprojects, such as

• Programme the Gauss algorithm over the integers and explore it.
• Solve systems of linear equations modulo a positive integer n.
• Find vectors with small integer entries that are orthogonal to a given set of vectors. Study some distribution/probability questions experimentally.

Supervisors: Gabor Wiese, Luca Notarnicola

Difficulty level: EML 1 (especially for students of the lecture Linear Algebra 2).

Tools: Any computer language.

Results: Report.