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Advanced Linear and Matrix Algebra
This textbook emphasizes the interplay between algebra and geometry to motivate the study of advanced linear algebra techniques. Matrices and linear transformations are presented as two sides of the same coin, with their connection motivating inquiry throughout the book. Building on a first course in linear algebra, this book offers readers a deeper understanding of abstract structures, matrix decompositions, multilinearity, and tensors. Concepts draw on concrete examples throughout, offering accessible pathways to advanced techniques.
Beginning with a study of vector spaces that includes coordinates, isomorphisms, orthogonality, and projections, the book goes on to focus on matrix decompositions. Numerous decompositions are explored, including the Shur, spectral, singular value, and Jordan decompositions. In each case, the author ties the new technique back to familiar ones, to create a coherent set of tools. Tensors and multilinearity complete the book, with a study of the Kronecker product, multilinear transformations, and tensor products. Throughout, “Extra Topic” sections augment the core content with a wide range of ideas and applications, from the QR and Cholesky decompositions, to matrix-valued linear maps and semidefinite programming. Exercises of all levels accompany each section.
Advanced Linear and Matrix Algebra offers students of mathematics, data analysis, and beyond the essential tools and concepts needed for further study. The engaging color presentation and frequent marginal notes showcase the author’s visual approach. A first course in proof-based linear algebra is assumed. An ideal preparation can be found in the author’s companion volume, Introduction to Linear and Matrix Algebra.
Beginning with a study of vector spaces that includes coordinates, isomorphisms, orthogonality, and projections, the book goes on to focus on matrix decompositions. Numerous decompositions are explored, including the Shur, spectral, singular value, and Jordan decompositions. In each case, the author ties the new technique back to familiar ones, to create a coherent set of tools. Tensors and multilinearity complete the book, with a study of the Kronecker product, multilinear transformations, and tensor products. Throughout, “Extra Topic” sections augment the core content with a wide range of ideas and applications, from the QR and Cholesky decompositions, to matrix-valued linear maps and semidefinite programming. Exercises of all levels accompany each section.
Advanced Linear and Matrix Algebra offers students of mathematics, data analysis, and beyond the essential tools and concepts needed for further study. The engaging color presentation and frequent marginal notes showcase the author’s visual approach. A first course in proof-based linear algebra is assumed. An ideal preparation can be found in the author’s companion volume, Introduction to Linear and Matrix Algebra.
- GenresMathematics
510 pages, Hardcover
Published May 20, 2021
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June 25, 2023
This is an extremely well-crafted book on linear algebra: from the quality of the paper, to the layout, and of course the content, there is nothing to complain about it.
It is written in a very pedagogical and clear way, with many examples and counter examples, which greatly help in understanding the covered topics.
The first part of the book is about linear spaces. The content comprises what is generally covered in a linear algebra book, with the "extra topic" section covering some topics which one generally does not find in such books, such as the QR decomposition.
The second part is where this book really starts shining. Here a complete introduction to matrix decompositions is given. Spectral, polar and singular value decompositions, as well as Schur triangularization and Jordan decomposition are covered in a clear way, with emphasis on the motivations that lead to their invention.
The third part of the book is also something generally not found in most introductory-to-intermediate books of linear algebra. Here multilinear transformations, their representations by multidimensional arrays and matrices, and tensor product spaces are introduced. The material covered in this part is at a higher level of abstraction than the rest of the book, but the author gives several concrete examples.
Overall the book is a real gem and highly recommended for who is interested intermediate-to-advanced linear algebra topics.
It is written in a very pedagogical and clear way, with many examples and counter examples, which greatly help in understanding the covered topics.
The first part of the book is about linear spaces. The content comprises what is generally covered in a linear algebra book, with the "extra topic" section covering some topics which one generally does not find in such books, such as the QR decomposition.
The second part is where this book really starts shining. Here a complete introduction to matrix decompositions is given. Spectral, polar and singular value decompositions, as well as Schur triangularization and Jordan decomposition are covered in a clear way, with emphasis on the motivations that lead to their invention.
The third part of the book is also something generally not found in most introductory-to-intermediate books of linear algebra. Here multilinear transformations, their representations by multidimensional arrays and matrices, and tensor product spaces are introduced. The material covered in this part is at a higher level of abstraction than the rest of the book, but the author gives several concrete examples.
Overall the book is a real gem and highly recommended for who is interested intermediate-to-advanced linear algebra topics.
Displaying 1 of 1 review