# S. Kumaresan's Linear Algebra: A Geometric Approach - A Must-Read for Mathematics Students

## Linear Algebra: A Geometric Approach by S. Kumaresan - An Introduction and Summary

Linear algebra is a branch of mathematics that studies the properties and operations of vectors, matrices, linear equations, linear transformations, vector spaces, and other related concepts. Linear algebra is widely used in various fields of science, engineering, and computer science, as it provides powerful tools for modeling and solving complex problems.

## linearalgebraageometricapproachbyskumaresan

However, many students find linear algebra abstract and difficult to understand, as it involves a lot of symbols, calculations, and proofs. Some students may also wonder what is the meaning and purpose of learning linear algebra, and how it relates to the real world.

This is where S. Kumaresan's book Linear Algebra: A Geometric Approach comes in handy. This book is designed for a first course in linear algebra and is intended for undergraduate courses in mathematics. It focuses throughout on geometric explanations to make the student perceive that linear algebra is nothing but analytic geometry of n dimensions. From the very start, linear algebra is presented as an extension of the theory of simultaneous linear equations and their geometric interpretation is shown to be a recurring theme of the subject. The integration of abstract algebraic concepts with the underlying geometric notions is one of the most distinguishing features of this book designed to help students in the pursuit of multivariable calculus and differential geometry in subsequent courses.

Explanations and concepts are logically presented in a conversational tone and well-constructed writing style so that students at a variety of levels can understand the material and acquire a solid foundation in the basic skills of linear algebra. The book also contains numerous examples, exercises, figures, and diagrams to illustrate and reinforce the concepts.

## What are the main topics covered in this book?

The book consists of eight chapters, each covering a major topic in linear algebra. Here is a brief overview of each chapter:

Chapter 1: Introduction: This chapter introduces the basic concepts and terminology of linear algebra, such as vectors, matrices, linear equations, linear transformations, vector spaces, subspaces, bases, dimension, rank, nullity, etc. It also shows how these concepts can be interpreted geometrically in two or three dimensions.

Chapter 2: Determinants: This chapter defines and explains the concept of determinants, which are useful for finding solutions of linear equations, inverse matrices, eigenvalues, etc. It also shows how determinants can be computed using various methods, such as cofactor expansion, row reduction, Laplace expansion, etc. It also shows how determinants can be used to measure areas and volumes.

Chapter 3: Eigenvalues and Eigenvectors: This chapter introduces and explains the concept of eigenvalues and eigenvectors, which are special values and vectors associated with a linear transformation or a matrix. It also shows how to find eigenvalues and eigenvectors using various methods, such as characteristic polynomial, Cayley-Hamilton theorem, diagonalization, etc. It also shows how eigenvalues and eigenvectors can be used to analyze various phenomena, such as stability, oscillation, rotation, etc.

Chapter 4: Inner Product Spaces: This chapter introduces and explains the concept of inner product spaces, which are vector spaces equipped with an additional operation called inner product or dot product. It also shows how inner product spaces can be used to define various notions such as length or norm, angle or cosine,

distance or metric,

orthogonality or perpendicularity,

projection or component,

orthonormal basis or unit vectors,

Gram-Schmidt process or orthogonalization,

etc. It also shows how inner product spaces can be used to study various topics such as least squares approximation,

Fourier series,

orthogonal matrices,

etc.

Chapter 5: Linear Functionals and Dual Spaces: This chapter introduces and explains the concept of linear functionals and dual spaces,

which are functions and spaces related to vector spaces by mapping vectors to scalars.

It also shows how linear functionals and dual spaces can be used to define various notions such as

annihilator or kernel,

transpose or adjoint,

dual basis or coordinate functions,

etc. It also shows how linear functionals and dual spaces can be used to study various topics such as

linear forms or equations,

bilinear forms or quadratic forms,

symmetric forms or matrices,

etc.

Chapter 6: Normed Linear Spaces: This chapter introduces and explains the concept of normed linear spaces,

which are vector spaces equipped with an additional operation called norm or length.

It also shows how normed linear spaces can be used to define various notions such as

convergence or limit,

continuity or smoothness,

differentiability or tangent,

etc. It also shows how normed linear spaces can be used to study various topics such as

Banach spaces or complete spaces,

Hilbert spaces or inner product spaces,

operators or functions,

etc.

Chapter 7: Linear Operators on Finite Dimensional Spaces: This chapter introduces and explains the concept of linear operators on finite dimensional spaces,

which are functions that map vectors from one vector space to another vector space while preserving linearity.

It also shows how linear operators on finite dimensional spaces can be used to define various notions such as

matrix representation or transformation matrix,

change of basis or similarity transformation,

invariant subspace or eigenspace,

etc. It also shows how linear operators on finite dimensional spaces can be used to study various topics such as

direct sum or decomposition,

Jordan canonical form or normal form,

spectral theorem or diagonalization theorem,

etc.

Chapter 8: Linear Operators on Infinite Dimensional Spaces: This chapter introduces and explains the concept of linear operators on infinite dimensional spaces,

which are functions that map vectors from one vector space to another vector space while preserving linearity.

It also shows how linear operators on infinite dimensional spaces can be used to define various notions such as

adjoint operator or Hermitian operator,

self-adjoint operator or symmetric operator,

positive operator or positive definite operator,

etc. It also shows how linear operators on infinite dimensional spaces can be used to study various topics such as

compact operator or finite rank operator,

Fredholm operator or index theory,

Spectral theorem for compact self-adjoint operators or eigenfunction expansion theorem,

etc.

## What are the main features of this book?

Besides the geometric approach and the integration of abstract and concrete concepts, this book has several other features that make it a valuable and enjoyable resource for learning linear algebra. Here are some of them:

The book is clear, concise, and highly readable. The author uses a conversational tone and a well-constructed writing style to explain the concepts and examples. The book avoids unnecessary jargon and technicalities and focuses on the essential ideas and techniques.

The book is logical and coherent. The author organizes the material in a logical and coherent manner, following a natural progression from simple to complex topics. The book also provides clear connections and transitions between different topics and chapters.

The book is comprehensive and rigorous. The author covers all the major topics and subtopics in linear algebra, providing complete and rigorous proofs and explanations. The book also includes some advanced topics that are usually not found in other introductory texts, such as dual spaces, normed linear spaces, linear operators on infinite dimensional spaces, etc.

The book is practical and relevant. The author shows how linear algebra can be applied to various fields of science, engineering, and computer science, such as geometry, calculus, differential equations, physics, chemistry, biology, cryptography, computer graphics, etc. The book also provides historical notes and biographical sketches of some of the prominent mathematicians who contributed to the development of linear algebra.

The book is interactive and engaging. The author encourages the reader to actively participate in the learning process by providing numerous examples, exercises, figures, and diagrams to illustrate and reinforce the concepts. The book also provides hints, answers, and solutions to some of the exercises at the end of each chapter.

## Who is the author of this book?

The author of this book is S. Kumaresan, a professor of mathematics at the University of Mumbai, India. He is the recipient of C.L.C. Chandna award for the year 1998 for excellence in mathematical teaching and research. He has been the Programme Director of Mathematics Training and Talent Search Programme for the last several years.

S. Kumaresan has a passion for teaching and writing mathematics. He has written several books and articles on various topics in mathematics, such as topology, analysis, algebra, geometry, etc. He has also been involved in various mathematical activities and initiatives, such as conducting workshops, seminars, lectures, competitions, etc. He has inspired and mentored many students and teachers in mathematics.

## How to get this book?

If you are interested in getting this book, you have several options to choose from. You can buy the book online or offline from various sources, such as Amazon, Flipkart, Google Books, etc. You can also borrow the book from a library or a friend. You can also download the book as a PDF file from some websites, such as Zlib.pub or Scribd.com.

However, before you get this book, you should be aware of some factors that may affect your experience with this book. For example, you should check the availability and price of the book from different sources and compare them to find the best deal. You should also check the quality and condition of the book if you buy or borrow it from a physical source. You should also check the legality and safety of the book if you download it from an online source.

### Conclusion

Linear Algebra: A Geometric Approach by S. Kumaresan is a book that aims to make linear algebra accessible and enjoyable for students and teachers of mathematics. It uses a geometric approach to explain the concepts and applications of linear algebra, showing that linear algebra is nothing but analytic geometry of n dimensions. It also integrates abstract and concrete concepts, providing a balance between theory and practice. It covers all the major topics and subtopics in linear algebra, providing clear, concise, and rigorous explanations and proofs. It also includes some advanced topics that are usually not found in other introductory texts. It provides numerous examples, exercises, figures, and diagrams to illustrate and reinforce the concepts. It also shows how linear algebra can be applied to various fields of science, engineering, and computer science. It also provides historical notes and biographical sketches of some of the prominent mathematicians who contributed to the development of linear algebra.

This book is suitable for a first course in linear algebra and is intended for undergraduate courses in mathematics. It can also be used as a reference or a supplementary text for higher level courses or self-study. It can help students and teachers in the pursuit of multivariable calculus and differential geometry in subsequent courses. It can also help anyone who wants to learn or improve their understanding of linear algebra.

If you are looking for a book that can make linear algebra easy and fun, you should definitely check out Linear Algebra: A Geometric Approach by S. Kumaresan. a27c54c0b2

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