Preface
Linear algebra is the source of many good things in this world. First of all, everything algebra, for sure. But also geometry and analysis, because any smooth function or manifold, taken locally, perturbes a certain linear transformation of [math]\mathbb R^N[/math]. And finally probability too, remember indeed that Gauss integral needed for talking about normal laws, which can only be computed by using polar coordinates and their Jacobian.
The purpose of this book is to talk about linear algebra in a large sense, theory and applications, at a somewhat more advanced level than the beginner one, and by insisting on beautiful things. And with some graduate level mathematics, and quantum physics too, in mind. We will particularly insist on the groups of matrices, which are extremely useful for all sorts of mathematics and physics, and which are perhaps the most beautiful topic one could study, once the basics of linear algebra and matrices understood.
The first half of the book is concerned with linear algebra and its applications. Part I is a quick journey through basic linear algebra, from basic definitions and fun with [math]2\times2[/math] matrices, up to the Spectral Theorem in its most general form, for the normal matrices [math]A\in M_N(\mathbb C)[/math]. Among the features of our presentation, the determinant will be introduced as it should, as a signed volume of a system of vectors. And also, we will discuss all sorts of useful matrix tricks, which are more advanced, and good to know.
As a continuation of this, Part II deals with various applications of linear algebra, to questions in analysis. After a quick look at differentiation and integration, which in several variables are intimately related to matrix theory, via the Jacobian, Hessian and so on, we will develop some useful probability theory, in relation with the normal and hyperspherical laws, by using spherical coordinates and their Jacobian. We will also discuss some other analytic topics, such as special matrices and spectral theory.
The second half of the book is concerned with matrix groups. As already mentioned, this is perhaps the most beautiful topic one could study, once the basics of linear algebra understood. The subject is however huge, and Part III will be a modest introduction to it. Our philosophy will be that of talking about all sorts of interesting closed subgroups [math]G\subset U_N[/math], finite and continuous alike, and by using very basic methods, coming from standard calculus, combinatorics and probability, for their study.
As a conclusion to this, the finite group case will appear to be reasonably understood, while the continuous case, not. Part IV will be dedicated to the study of the closed subgroups [math]G\subset U_N[/math], and more specifically the continuous ones, by using heavy machinery, as heavy as it gets. We will discuss here the basics of representation theory, then the existence of the Haar measure, and the Peter-Weyl theory, and then more advanced topics, such as Tannakian duality, Brauer theorems, and Weingarten calculus.
In the hope that you will find this book useful. At the level of things which are not done here, notable topics include the Jordan decomposition, which is the nightmare of everyone involved, teacher or student, and this remains between us, as well as some basic Lie algebra theory, which would have perfectly make sense to include, but that we preferred to replace by representation theory, and its relation with combinatorics and probability, which are somewhat more elementary, and fitting better with the rest.
Finally, let us mention that this way of presenting things has its origins in some recent research work on the quantum groups, and more specifically on the so-called easy quantum groups. The idea there is that there is no much smoothness and geometry, with the main tools belonging to combinatorics and probability. Thus, as main philosophy, the present book, while dealing with classical topics, is written with a “quantum” touch.
Most of this book is based on lecture notes from various classes at Cergy, and I would like to thank my students. The final part goes into research topics, and I am grateful to Beno\^it Collins, Steve Curran and Jean-Marc Schlenker, for our joint work on the subject. Many thanks go as well to my cats. There is so much to learn from them, too.
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Cergy, July 2024
Teo Banica \baselineskip=15.95pt \tableofcontents \baselineskip=14pt
General references
Banica, Teo (2024). "Linear algebra and group theory". arXiv:2206.09283 [math.CO].