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2019 | Buch

Linear Algebra for Computational Sciences and Engineering

verfasst von: Assoc. Prof. Ferrante Neri

Verlag: Springer International Publishing

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

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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.

Inhaltsverzeichnis

Frontmatter

Foundations of Linear Algebra

Frontmatter

Chapter 1. Basic Mathematical Thinking

Abstract

Mathematics, from the Greek word “mathema”, is simply translated as science or expression of the knowledge. Regardless of the fact that mathematics is something that exists in our brain as well as in the surrounding nature and we discover it little by little or an invention/abstraction of a human brain, mathematics has been with us with our capability of thinking and is the engine of human progress.

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Chapter 2. Matrices

Abstract

Although this chapter intentionally refers to the set of real numbers \(\mathbb {R}\) and its sum and multiplication operations, all the concepts contained in this chapter can be easily extended to the set of complex numbers \(\mathbb {C}\) and the complex field. This fact is further remarked in Chap. 5 after complex numbers and their operations are introduced.

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Chapter 3. Systems of Linear Equations

Abstract

A linear equation in \(\mathbb {R}\) in the variables x ₁, x ₂, …, x _n is an equation of the kind

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Chapter 4. Geometric Vectors

Abstract

It can be proved that \(\mathbb {R}\) is a continuous 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

Abstract

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. For example it is easy to verify that \(\mathbb {R}\) is closed with respect to the sum as the sum of two real numbers is certainly a real number. On the other hand, \(\mathbb {R}\) is not closed with respect to the square root operation.

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Chapter 6. An Introduction to Geometric Algebra and Conics

Abstract

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

Frontmatter

Chapter 7. An Overview on Algebraic Structures

Abstract

This chapter recaps and formalizes concepts used in the previous sections of this book. Furthermore, this chapter reorganizes and describes in depth the topics mentioned at the end of Chap. 1, i.e. a formal characterization of the abstract algebraic structures and their hierarchy. This chapter is thus a revisited summary of concepts previously introduced and used and provides the mathematical basis for the following chapters.

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Chapter 8. Vector Spaces

Abstract

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. An Introduction to Inner Product Spaces: Euclidean Spaces

Abstract

Let \(\left (E,+,\cdot \right )\) be a finite-dimensional vector space over the field \(\mathbb {K}\).

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Chapter 10. Linear Mappings

Abstract

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. More specifically, while the origin of the term algebra has been mentioned in Chap. 1, the use of the adjective linear has never been discussed. Before entering into the formal definitions of linearity, let us illustrate the subject at the intuitive level. Linear algebra can be seen as a subject that studies vectors. If we consider that vector spaces are still vectors endowed with composition laws, that matrices are collections of row (or column) vectors, that systems of linear equations are vector equations, and that a number can be interpreted as a single element vector, we see that the concept of vector is the elementary entity of linear algebra. As seen in Chap. 4, a vector is generated by a segment of a line. Hence, the subject linear algebra studies “portions” of lines and their interactions, which justifies the adjective “linear”.

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Chapter 11. An Introduction to Computational Complexity

Abstract

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. More specifically, this chapter gives some basics of complexity theory and discrete mathematics and will attempt to answer to the question: “What is a hard problem?”

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Chapter 12. Graph Theory

Abstract

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 13. Applied Linear Algebra: Electrical Networks

Abstract

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 everyday 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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Backmatter

Titel: Linear Algebra for Computational Sciences and Engineering
verfasst von: Assoc. Prof. Ferrante Neri
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-21321-3
Print ISBN: 978-3-030-21320-6
DOI: https://doi.org/10.1007/978-3-030-21321-3