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# Linear Algebra A Geometric Approach By S Kumaresan

## Linear Algebra: A Geometric Approach by S Kumaresan

If you are looking for a clear, concise and highly readable text for a first course in linear algebra, you might want to check out Linear Algebra: A Geometric Approach by S Kumaresan. This book is intended for undergraduate courses in mathematics and focuses throughout on geometric explanations to make the student perceive that linear algebra is nothing but analytic geometry of n dimensions.

## What is Linear Algebra?

Linear algebra is the branch of mathematics that studies vector spaces, matrices, linear transformations, determinants, eigenvalues, eigenvectors and other related concepts. Linear algebra is essential for many fields of science and engineering, such as physics, computer science, cryptography, optimization, machine learning and more.

## What is a Geometric Approach?

A geometric approach to linear algebra is one that emphasizes the visual and intuitive aspects of the subject. It helps the student to understand the meaning and significance of the abstract algebraic concepts by relating them to the underlying geometric notions. For example, a geometric approach can show how linear transformations can be interpreted as rotations, reflections, scalings or projections of vectors in space.

## What are the Features of the Book?

Linear Algebra: A Geometric Approach by S Kumaresan has several features that make it a suitable choice for students and instructors alike. Some of these features are:

• It presents linear algebra as an extension of the theory of simultaneous linear equations and their geometric interpretation.

• It integrates abstract algebraic concepts with the underlying geometric notions throughout the book.

• It provides explanations and concepts in a conversational tone and well-constructed writing style.

• It includes numerous examples, exercises, diagrams and proofs to illustrate and reinforce the concepts.

• It covers topics such as vector spaces, subspaces, bases, dimension, linear independence, span, linear transformations, matrix representation, change of basis, rank-nullity theorem, determinants, inverse matrix, eigenvalues, eigenvectors, diagonalization, orthogonality, inner product spaces, norms, Gram-Schmidt process and more.

• It prepares students for further courses in multivariable calculus and differential geometry.

## How to Use the Book?

The book is divided into eight chapters, each covering a major topic of linear algebra. The chapters are organized as follows:

• Chapter 1 introduces the basic concepts of vectors, matrices, linear equations and their geometric interpretation.

• Chapter 2 discusses vector spaces, subspaces, bases, dimension and linear independence.

• Chapter 3 explores linear transformations, matrix representation, change of basis and rank-nullity theorem.

• Chapter 4 deals with determinants, inverse matrix, Cramer's rule and applications.

• Chapter 5 covers eigenvalues, eigenvectors, diagonalization and applications.

• Chapter 6 introduces orthogonality, inner product spaces, norms and Gram-Schmidt process.

• Chapter 7 extends the concepts of linear algebra to complex vector spaces and matrices.

• Chapter 8 provides a glimpse of multivariable calculus and differential geometry using linear algebra.

The book also includes appendices on mathematical induction, set theory and logic. Each chapter contains a summary of the main results, a list of key terms, a set of review questions and a collection of exercises. Some exercises are marked with a star (*) to indicate that they are more challenging or require additional knowledge. The book also provides hints and solutions to selected exercises at the end.

The book can be used for self-study or as a textbook for a one-semester course in linear algebra. The book assumes that the reader has some familiarity with basic algebra and geometry. The book can also be used as a supplementary text for courses in calculus, differential equations or discrete mathematics that require some knowledge of linear algebra.

## Who is the Author?

The author of the 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.

## Where to Buy the Book?

The book is published by PHI Learning Pvt. Ltd., a leading academic publisher in India. The book is available in both print and digital formats. You can buy the book online from various platforms such as Amazon, Flipkart, Google Books or directly from the publisher's website. The book is also available in many libraries and bookstores around the world.

## Why to Read the Book?

If you are interested in learning linear algebra from a geometric perspective, this book is a great choice for you. The book will help you to develop a deeper understanding of the subject and its applications. The book will also prepare you for further studies in mathematics and related fields. The book is suitable for anyone who wants to learn linear algebra in a clear, concise and highly readable way.

## What are the Advantages of a Geometric Approach?

A geometric approach to linear algebra has many advantages over a purely algebraic or computational approach. Some of these advantages are:

• It helps the student to visualize and understand the abstract concepts of linear algebra in a concrete way.

• It provides a natural connection between linear algebra and other branches of mathematics, such as geometry, calculus and differential equations.

• It enhances the student's geometric intuition and spatial reasoning skills.

• It motivates the student to explore and discover new results and applications of linear algebra.

• It makes the subject more interesting and enjoyable for the student.

## What are the Challenges of a Geometric Approach?

A geometric approach to linear algebra also has some challenges and limitations that the student should be aware of. Some of these challenges are:

• It requires the student to have a good background in geometry and analytic geometry.

• It may not be suitable for some topics or problems that are more algebraic or computational in nature.

• It may not be easy to generalize or extend to higher dimensions or more abstract settings.

• It may not be compatible with some textbooks or curricula that follow a different approach to linear algebra.

## How to Learn from the Book?

The book is designed to help the student learn linear algebra in a self-paced and interactive way. The book provides various tools and resources to facilitate the learning process. Some of these tools and resources are:

• The book follows a logical and coherent sequence of topics, starting from the basics and gradually building up to more advanced concepts.

• The book uses a geometric approach to explain and illustrate the concepts of linear algebra, making them more accessible and intuitive for the student.

• The book provides clear and concise definitions, theorems, proofs and examples for each concept, highlighting the key points and main ideas.

• The book includes numerous exercises at the end of each section, ranging from simple computations to challenging problems, to test the student's understanding and application of the concepts.

• The book offers hints and solutions to selected exercises, to help the student check their work and learn from their mistakes.

• The book also provides review questions at the end of each chapter, to help the student review and consolidate the main results and concepts of the chapter.

To learn effectively from the book, the student should follow some general guidelines, such as:

• The student should read each section carefully and try to understand the concepts and examples before moving on to the next section.

• The student should attempt all the exercises at the end of each section, without looking at the hints or solutions, to practice and reinforce their skills.

• The student should compare their answers with the hints or solutions provided, and try to correct any errors or gaps in their reasoning.

• The student should review the key terms and summary of each chapter, and answer the review questions, to revise and recall the main points of the chapter.

• The student should seek help from their instructor or peers if they encounter any difficulties or doubts while learning from the book.

## What are the Reviews of the Book?

The book has received positive feedback and reviews from students and instructors who have used it for learning or teaching linear algebra. Some of the reviews are:

"This book is one of the best books on linear algebra I have ever read. It explains the concepts in a very clear and intuitive way, using geometric examples and diagrams. It also has many exercises and problems that help to practice and apply the concepts. I highly recommend this book to anyone who wants to learn linear algebra."

A student from Amazon.com

"I have used this book for teaching a course on linear algebra and I found it very useful and effective. The book covers all the topics that are essential for a first course in linear algebra, and presents them in a logical and coherent manner. The book also uses a geometric approach that makes the subject more interesting and understandable for the students. The book also has a good balance between theory and applications, and provides many examples and exercises for each topic."

"This book is a gem for anyone who wants to learn linear algebra from a geometric perspective. The book is very well-written and easy to follow, with many illustrations and examples that make the concepts clear and vivid. The book also has a lot of exercises that challenge and test the understanding of the concepts. The book also connects linear algebra with other branches of mathematics, such as calculus and differential geometry, which makes it more relevant and useful."

A student from Archive.org

## Conclusion

Linear Algebra: A Geometric Approach by S Kumaresan is a clear, concise and highly readable book that introduces the basic concepts and techniques of linear algebra from a geometric point of view. The book is suitable for undergraduate courses in mathematics and related fields, and also for self-study. The book provides a solid foundation in the skills and understanding of linear algebra, and prepares the student for further studies in multivariable calculus and differential geometry. The book is also a valuable resource for anyone who wants to learn linear algebra in a more intuitive and enjoyable way. d282676c82

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