Techniques in Linear Algebra workshop

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A one day workshop to help bring together early stage researchers to learn and discuss topics in Linear algebra.

Linear algebra is frequently introduced to first or second year undergraduates, and afterwards its results and concepts are taken for granted. But it is the rare undergraduate who understands the full details of the Jordan Canonical Form at the first presentation. In this short course, we will devote one morning to a complete analysis of a linear operator in terms of generalised eigenvalues. We will explain the significance of the Cayley-Hamilton theorem (and why no-one states the result appropriately). In the end, we will have a thorough understanding of the structure underlying the Jordan Form. Our approach will be mostly elementary, though we may use the word 'module' on occasion. 

In a second session, we will discuss the classification of quadratic forms, with particular attention to the fields Q, R, C. This generalises the theory of inner product spaces (which again, is often introduced but rarely mastered at undergraduate level). We will discuss the polarisation of forms, the classification by Hilbert symbols and state the Local-Global principal. We focus on concrete computations - knowledge of p-adic numbers will not be required! As an application, we will give a complete and self-contained proof of the Bruck-Ryser-Chowla theorem, ruling out the existence of certain projective planes. 

This workshop is suitable for advanced undergraduate students and graduate students at any career stage. It may also be of interest to colleagues looking to refresh their knowledge of these classical algebraic topics. The only pre-requisite is a thorough understanding of elementary linear algebra: conceptual understanding of diagonalisation of a matrix should suffice.

 To help with catering and numbers, please register if you plan on attending, there is no registration fee.

For any help or extra details please email:

patrick.browne@tus.ie