A Review of Fundamentals of Linear Algebra by Katsumi Nomizu
Fundamentals of Linear Algebra is a classic textbook on the subject of linear algebra, written by Katsumi Nomizu, a Japanese mathematician and professor at Brown University. The book was first published in 1966 by McGraw-Hill, and later reprinted in 1979 by Chelsea Publishing Company. The book covers the basic concepts and techniques of linear algebra, such as vector spaces, linear mappings, matrices, determinants, minimal polynomials, inner products, affine spaces, and euclidean spaces. The book also introduces some algebraic concepts, such as ideals, fields, and characteristic polynomials. The book is suitable for undergraduate students who have some background in calculus and abstract algebra.
The book is divided into 11 chapters, each with a number of exercises and examples. The first chapter introduces the notion of a vector space over a field F, and discusses some properties and examples of vector spaces. The second chapter studies linear mappings and matrices, and shows how to represent linear mappings by matrices with respect to different bases. The third chapter deals with the construction of vector spaces from other vector spaces, such as direct sums, quotients, dual spaces, and tensor products. The fourth chapter reviews some algebraic concepts that are useful for linear algebra, such as rings, ideals, fields, homomorphisms, isomorphisms, and quotient rings. The fifth chapter introduces determinants and their properties, such as Laplace expansion, Cramer's rule, and Cayley-Hamilton theorem.
The sixth chapter focuses on minimal polynomials and characteristic polynomials of linear mappings and matrices. It also discusses the Jordan canonical form and rational canonical form of matrices. The seventh chapter introduces inner product spaces and their properties, such as orthogonality, norms, angles, Cauchy-Schwarz inequality, Gram-Schmidt process, and orthogonal complements. The eighth chapter studies affine spaces and their properties, such as affine combinations, barycentric coordinates, affine transformations, affine subspaces, and affine frames. The ninth chapter explores euclidean spaces and their properties, such as distances, angles, isometries, rotations, reflections, orthogonal matrices, unitary matrices, and orthonormal bases.
The tenth chapter is an appendix that contains some useful results from analysis and algebra that are used in the book. For example, it proves the existence and uniqueness of solutions for systems of linear equations using the contraction mapping principle. It also proves some results about roots of polynomials using the fundamental theorem of algebra. The eleventh chapter is a list of symbols used in the book.
The book is well-written and rigorous. It provides clear definitions and proofs for all the main results. It also gives many examples and exercises to illustrate the concepts and techniques. The book covers most of the topics that are usually taught in a standard linear algebra course. However, some topics that are often included in modern linear algebra courses are missing from the book. For example,
the book does not discuss eigenvalues and eigenvectors of linear mappings and matrices,
the book does not cover topics related to diagonalization of matrices or diagonalizable linear mappings,
the book does not introduce topics related to bilinear forms or quadratic forms,
the book does not mention topics related to spectral theory or singular value decomposition.
These topics may be covered in more advanced courses on linear algebra or matrix analysis.
Fundamentals of Linear Algebra by Katsumi Nomizu is a classic textbook on linear algebra that covers the basic concepts and techniques of the subject. It is suitable for undergraduate students who have some background in calculus and abstract algebra. However,
the book may be outdated for some modern applications of linear algebra,
the book may be too difficult for some students who are not familiar with abstract algebra,
the book may be too dry for some students who prefer more visual or intuitive explanations.
the book may not be the best choice for everyone who wants to learn linear algebra. ec8f644aee