Braids and linear algebra

A Linear Ordering of Braids

In this chapter, we introduce the linear ordering of braids, sometimes called the Dehornoy ordering, that is the main subject of this book, and we list its main properties known so far. The construction starts with the notion of a σ-positive braid, and it relies on three basic properties, called A, C, and S, from which the σ-ordering can easily be constructed and investigated. In this chapter, ...

Trigonometric Identities, Linear Algebra, and Computer Algebra

The Algebra of Linear Algebra

3. You are already familiar with the concept of vector spaces in the form of …nite dimensional Euclidean vector spaces, and most of what we do in these lectures will involve these spaces (and, indeed, will also assume that the dot product operation is well-de…ned so that we can de…ne the matrix operations associated with linear transformations). However, it will also be useful to consider some ...

Linear and Multilinear Algebra

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Dynamics and Linear Algebra

This course provides an introduction to the interplay between linear algebra and di¤erential equations/dynamical systems in continuous time. We …rst introduce linear di¤erential equations in Euclidian space R and on Grassmannian and ‡ag manifolds induced by a single matrix A, with emphasis on characterizations of the constant matrix A from a dynamics point of view. We then introduce linear skew...