nach oben

2011 | Buch

Kapitel lesen Erstes Kapitel lesen

Linear algebra

for everyone

verfasst von: Lorenzo Robbiano

Verlag: Springer Milan

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten

Über dieses Buch

This book provides students with the rudiments of Linear Algebra, a fundamental subject for students in all areas of science and technology. The book would also be good for statistics students studying linear algebra.

It is the translation of a successful textbook currently being used in Italy. The author is a mathematician sensitive to the needs of a general audience.

In addition to introducing fundamental ideas in Linear Algebra through a wide variety of interesting examples, the book also discusses topics not usually covered in an elementary text (e.g. the "cost" of operations, generalized inverses, approximate solutions). The challenge is to show why the "everyone" in the title can find Linear Algebra useful and easy to learn.

The translation has been prepared by a native English speaking mathematician, Professor Anthony V. Geramita.

Inhaltsverzeichnis

Frontmatter

Numerical and Symbolic Computations

Abstract

Suppose that a reader, perhaps intrigued by the title, wanted to see immediately if the book was really for everyone and consequently arrived here without having read neither the Forward nor the Introduction. I believe that such a reader would have made an error and missed an essential aspect of the spirit of the book. I would strongly advise such a reader to return and read those parts of the book. However, since the reader is at liberty to do as he or she chooses, and since I personally know many readers who have the habit (may I say the bad habit?) of not reading introductions, I have decided not to mislead even those readers and so I will begin with this very short chapter, a typical Chapter 0, in which one does some simple computations and then discusses the results obtained by those computations.

Lorenzo Robbiano

Part I

Frontmatter

1. Systems of Linear Equations and Matrices

Abstract

In the introductory chapter we warmed up the engines by studying the equation ax = b. What will we do to follow up on that? I will tell you right away that in this chapter we will deal with transportation problems and chemical reactions, diet manipulations, totocalcio tickets, architectural constructions and meteorology. How is that possible? Am I changing the scope of the book? On the contrary. The fascination of even not very sophisticated mathematics lies precisely in its ability to bring together topics which, at the outset, seem very different.

Lorenzo Robbiano

2. Operations with Matrices

Abstract

Matrix, matrices... how many times have we used these words. It probably won’t surprise you that we will continue to use those words frequently. The matrix is one of the most useful mathematical objects we have at our disposal, a basic tool for those who use mathematics. This is the case for lots of good reasons, some of which we have already seen and some we will see shortly.

Lorenzo Robbiano

3. Solutions of Systems of Linear Equations

Abstract

In this chapter we come to grips with the question of how to solve, in practice, systems of linear equations. Our strategy will be to gather a certain number of observations which will allow us to form a strategy. Given that many mathematicians often use the adjective clear for something which usually is far from clear but is tiresome to prove we will prove our good intention to avoid this bad habit by beginning with an observation that is totally clear.

Lorenzo Robbiano

4. Coordinate Systems

Abstract

Up till now we have spoken about algebraic objects, usually matrices and vectors. But, at the beginning of this book (see Section 1.2) we gave examples of vectors such as force, velocity, acceleration, coming from physics. It’s clear that we’ve used the word vector to mean at least two distinct things. But how can physical or geometric objects have the same name as purely algebraic ones? In this chapter we will study the how and the why and we will discover an extraordinary activity of the mathematical arts: the construction of models not only of physical, biological and statistical objects but also of other mathematical objects. Said another way: mathematical entities, often created to be models of something else, can themselves be modeled inside mathematics.

Lorenzo Robbiano

Part II

Frontmatter

5. Quadratic Forms

Abstract

In this chapter we will study yet another aspect of the extraordinary capacity of matrices to adapt themselves to very diverse situations.

Lorenzo Robbiano

6. Orthogonality and Orthonormality

Abstract

We are used to thinking of orthogonal coordinate systems as the most interesting and most useful. This habit comes from studying the graphs of functions in the plane or in space. Is the same thing also true in ℝⁿ? In this chapter we will try to justify the reason for this perception and also to respond to the question above.

Lorenzo Robbiano

7. Projections, Pseudoinverses and Least Squares

Abstract

The farmers would say that we are fast approaching the harvest season. We wouldn’t want to disillusion them and so we will begin (in this section) to harvest some of the fruit generated by the work of the previous sections. In particular we will discover how to solve, in various ways, a very important problem which goes by the name of the problem of least squares. But, we have to first equip ourselves with some important tools, namely linear transformations.

Lorenzo Robbiano

8. Endomorphisms and Diagonalization

Abstract

In this chapter we will study a central topic in the theory of matrices. We have already seen (and I hope we haven’t forgotten) that matrices serve as containers of numerical information, for example as the fundamental parts of linear systems and as an essential mathematical tool for their solution. We’ve multiplied them, we’ve calculated their inverses (when they exist) and we have decomposed them into the LU decomposition. Then we began to appreciate them as geometric tools, in the sense that we have associated them to vectors and systems of coordinates.

Lorenzo Robbiano

Backmatter

Titel: Linear algebra
verfasst von: Lorenzo Robbiano
Verlag: Springer Milan
Electronic ISBN: 978-88-470-1839-6
Print ISBN: 978-88-470-1838-9
DOI: https://doi.org/10.1007/978-88-470-1839-6