This book covers the basic theory of matrices and vector spaces. The book's three main parts cover (I) matrices, vector spaces, bases, and dimension; (II) inner products, bilinear and sesquilinear forms over vector spaces; (III) linear transformations, eigenvalues and eigenvectors, diagonalization, and Jordan normal form. An introduction to fields and polynomials over fields is also provided, and examples and applications are provided throughout. The approach
throughout is rigorous, but without being unnecessarily abstract. In particular, this book would be suitable reading for a student with no prior exposure to abstract algebra. Although intended as a
'second course', the book is completely self-contained and all the material usually given in a 'first course' in presented fully in Part I, so the book provides a useful guide to the entire theory of vector spaces as usually studied in an undergraduate degree. Abstract methods are illustrated with concrete examples throughout, and more detailed examples highlight applications of linear algebra to analysis, geometry, differential equations, relativity and quantum mechanics. As such, the book
provides a valuable introduction to a wide variety of mathematical methods.
This book covers the basic theory of matrices and vector spaces. The book's three main parts cover (I) matrices, vector spaces, bases, and dimension; (II) inner products, bilinear and sesquilinear forms over vector spaces; (III) linear transformations, eigenvalues and eigenvectors, diagonalization, and Jordan normal form. An introduction to fields and polynomials over fields is also provided, and examples and applications are provided throughout. The approach
throughout is rigorous, but without being unnecessarily abstract. In particular, this book would be suitable reading for a student with no prior exposure to abstract algebra. Although intended as a
'second course', the book is completely self-contained and all the material usually given in a 'first course' in presented fully in Part I, so the book provides a useful guide to the entire theory of vector spaces as usually studied in an undergraduate degree. Abstract methods are illustrated with concrete examples throughout, and more detailed examples highlight applications of linear algebra to analysis, geometry, differential equations, relativity and quantum mechanics. As such, the book
provides a valuable introduction to a wide variety of mathematical methods.
1: Matrices
2: Vector spaces
3: Inner product spaces
4: Bilinear and sesquilinear forms
5: Orthogonal bases
6: When in a form definite?
7: Quadratic forms
8: Linear transformations
9: Polynomials
10: Eigenvalues and eigenvectors
11: The minimum polynomial
12: Diagonalization
13: Self-adjoint transformations
14: The Jordan normal form
"Kaye offers this work as a second course in linear algebra. As such, it deals with the specific subject matter of linear algebra in a way that could also be viewed as an introduction to abstract algebra or axiomatic mathematics in general. Knowledge of elementary matrix arithmetic and matrix methods--including the general solution to systems of linear equations and computation of inverses and determinants--is assumed, though these topics are briefly reviewed. Some exposure to abstract vector spaces and the notions of basis and dimension would also be helpful to one wishing to peruse this book. For those with a suitable background, this book provides a very rigorous treatment of the fundamentals of linear algebra, including inner product spaces, bilinear and quadratic forms, orthogonal bases, eigenvalues and eigenvectors, and the Jordan canonical form. Certainly appropriate for upper-division undergraduates entertaining thoughts of graduate work in mathematics."--Choice
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