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Über dieses Buch

This book presents the main concepts of linear algebra from the viewpoint of applied scientists such as computer scientists and engineers, without compromising on mathematical rigor. Based on the idea that computational scientists and engineers need, in both research and professional life, an understanding of theoretical concepts of mathematics in order to be able to propose research advances and innovative solutions, every concept is thoroughly introduced and is accompanied by its informal interpretation. Furthermore, most of the theorems included are first rigorously proved and then shown in practice by a numerical example. When appropriate, topics are presented also by means of pseudocodes, thus highlighting the computer implementation of algebraic theory.
It is structured to be accessible to everybody, from students of pure mathematics who are approaching algebra for the first time to researchers and graduate students in applied sciences who need a theoretical manual of algebra to successfully perform their research. Most importantly, this book is designed to be ideal for both theoretical and practical minds and to offer to both alternative and complementary perspectives to study and understand linear algebra.



Foundations of Linear Algebra


Chapter 1. Basic Mathematical Thinking

Mathematics, from the Greek word “mathema”, is simply translated as science or expression of the knowledge.
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Chapter 2. Matrices

Let \(\mathbb {R}\) be the set of real number.
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Chapter 3. Systems of Linear Equations

A linear equation in \(\mathbb {R}\) in the variables \(x_1,x_2,\ldots ,x_n\) is an equation of the kind:
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Chapter 4. Geometric Vectors

It can be proved that \(\mathbb {R}\) is a dense set. As such, it can be graphically represented as an infinite continuous line, see [1].
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Chapter 5. Complex Numbers and Polynomials

As mentioned in Chap. 1, for a given set and an operator applied to its elements, if the result of the operation is still an element of the set regardless of the input of the operator, then the set is said closed with respect to that operator.
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Chapter 6. An Introduction to Geometric Algebra and Conics

This chapter introduces the conics and characterizes them from an algebraic perspective. While in depth geometrical aspects of the conics lie outside the scopes of this chapter, this chapter is an opportunity to revisit concepts studied in other chapters such as matrix and determinant and assign a new geometric characterization to them.
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Elements of Linear Algebra


Chapter 7. An Overview on Algebraic Structures

This chapter recaps and formalizes concepts used in the previous sections of this book.
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Chapter 8. Vector Spaces

This chapter revisits the concept of vector bringing it to an abstract level. Throughout this chapter, for analogy we will refer to vectors using the same notation as for numeric vectors.
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Chapter 9. Linear Mappings

Although the majority of the topics in this book (all the topics taken into account excluding only complex polynomials) are related to linear algebra, the subject “linear algebra” has never been introduced in the previous chapters.
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Chapter 10. An Introduction to Computational Complexity

This chapter is not strictly about algebra. However, this chapter offers a set of mathematical and computational instruments that will allow us to introduce several concepts in the following chapters. Moreover, the contents of this chapter are related to algebra as they are ancillary concepts that help (and in some cases allow) the understanding of algebra.
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Chapter 11. Graph Theory

In this chapter we introduce a notion of fundamental importance for modelling in schematic way a large amount of problems. This is the concept of a graph. This concept applies not only to computer science and mathematics, but even in fields as diverse as chemistry, biology, physics, civil engineering, mapping, telephone networks, electrical circuits, operational research, sociology, industrial organization, the theory of transport, artificial intelligence.
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Chapter 12. Applied Linear Algebra: Electrical Networks

This chapter shows how mathematical theory is not an abstract subject which has no connection with the real world. On the contrary, this entire book is written by stating that mathematics in general, and algebra in this case, is an integrating part of every day real life and that the professional life of computational scientists and engineers requires a solid mathematical background. In order to show how the contents of the previous chapters have an immediate technical application, the last chapter of this book describes a core engineering subject, i.e. electrical networks, as an algebraic exercise. Furthermore, this chapter shows how the combination of the algebraic topics give a natural representation of a set of interacting physical phenomena.
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