Ebook: Linear Algebra

This popular and successful text was originally written for a one- semester course in linear algebra at the sophomore undergraduate level. Students at this level generally have had little contact with complex numbers or abstract mathematics, so the book deals almost exclusively with real finite dimensional vector spaces, but in a setting and formulation that permits easy generalization to abstract vector spaces. The goal of the first two editions was the principal axis theorem for real symmetric linear transformation. The principal axis theorem becomes the first of two goals for this new edition, which follows a straight path to its solution. A wide selection of examples of vector spaces and linear transformation is presented to serve as a testing ground for the theory. In the second edition, a new chapter on Jordan normal form was added which reappears here in expanded form as the second goal of this new edition, along with applications to differential systems. To achieve the principal axis theorem in one semester a straight path to these two goals is followed. As compensation, there is a wide selection of examples and exercises. In addition, the author includes an introduction to invariant theory to show students that linear algebra alone is not capable of solving these canonical forms problems. The book continues to offer a compact, but mathematically clean introduction to linear algebra with particular emphasis on topics that are used in abstract algebra, the theory of differential equations, and group representation theory.

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This popular and successful text was originally written for a one- semester course in linear algebra at the sophomore undergraduate level. Students at this level generally have had little contact with complex numbers or abstract mathematics, so the book deals almost exclusively with real finite dimensional vector spaces, but in a setting and formulation that permits easy generalization to abstract vector spaces. The goal of the first two editions was the principal axis theorem for real symmetric linear transformation. The principal axis theorem becomes the first of two goals for this new edition, which follows a straight path to its solution. A wide selection of examples of vector spaces and linear transformation is presented to serve as a testing ground for the theory. In the second edition, a new chapter on Jordan normal form was added which reappears here in expanded form as the second goal of this new edition, along with applications to differential systems. To achieve the principal axis theorem in one semester a straight path to these two goals is followed. As compensation, there is a wide selection of examples and exercises. In addition, the author includes an introduction to invariant theory to show students that linear algebra alone is not capable of solving these canonical forms problems. The book continues to offer a compact, but mathematically clean introduction to linear algebra with particular emphasis on topics that are used in abstract algebra, the theory of differential equations, and group representation theory.
Content:
Front Matter....Pages i-xii
Vectors in the Plane and in Space....Pages 1-14
Vector Spaces....Pages 15-23
Examples of Vector Spaces....Pages 25-33
Subspaces....Pages 35-45
Linear Independence and Dependence....Pages 47-56
Finite-Dimensional Vector Spaces and Bases....Pages 57-74
The Elements of Vector Spaces: A Summing Up....Pages 75-83
Linear Transformations....Pages 85-112
Linear Transformations: Examples and Applications....Pages 113-128
Linear Transformations and Matrices....Pages 129-157
Representing Linear Transformations by Matrices....Pages 159-183
More on Representing Linear Transformations by Matrices....Pages 185-198
Systems of Linear Equations....Pages 199-226
The Elements of Eigenvalue and Eigenvector Theory....Pages 227-265
Inner Product Spaces....Pages 267-306
The Spectral Theorem and Quadratic Forms....Pages 307-341
Jordan Canonical Form....Pages 343-380
Application to Differential Equations....Pages 381-404
The Similarity Problem....Pages 405-410
Back Matter....Pages 411-454

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This popular and successful text was originally written for a one- semester course in linear algebra at the sophomore undergraduate level. Students at this level generally have had little contact with complex numbers or abstract mathematics, so the book deals almost exclusively with real finite dimensional vector spaces, but in a setting and formulation that permits easy generalization to abstract vector spaces. The goal of the first two editions was the principal axis theorem for real symmetric linear transformation. The principal axis theorem becomes the first of two goals for this new edition, which follows a straight path to its solution. A wide selection of examples of vector spaces and linear transformation is presented to serve as a testing ground for the theory. In the second edition, a new chapter on Jordan normal form was added which reappears here in expanded form as the second goal of this new edition, along with applications to differential systems. To achieve the principal axis theorem in one semester a straight path to these two goals is followed. As compensation, there is a wide selection of examples and exercises. In addition, the author includes an introduction to invariant theory to show students that linear algebra alone is not capable of solving these canonical forms problems. The book continues to offer a compact, but mathematically clean introduction to linear algebra with particular emphasis on topics that are used in abstract algebra, the theory of differential equations, and group representation theory.
Content:
Front Matter....Pages i-xii
Vectors in the Plane and in Space....Pages 1-14
Vector Spaces....Pages 15-23
Examples of Vector Spaces....Pages 25-33
Subspaces....Pages 35-45
Linear Independence and Dependence....Pages 47-56
Finite-Dimensional Vector Spaces and Bases....Pages 57-74
The Elements of Vector Spaces: A Summing Up....Pages 75-83
Linear Transformations....Pages 85-112
Linear Transformations: Examples and Applications....Pages 113-128
Linear Transformations and Matrices....Pages 129-157
Representing Linear Transformations by Matrices....Pages 159-183
More on Representing Linear Transformations by Matrices....Pages 185-198
Systems of Linear Equations....Pages 199-226
The Elements of Eigenvalue and Eigenvector Theory....Pages 227-265
Inner Product Spaces....Pages 267-306
The Spectral Theorem and Quadratic Forms....Pages 307-341
Jordan Canonical Form....Pages 343-380
Application to Differential Equations....Pages 381-404
The Similarity Problem....Pages 405-410
Back Matter....Pages 411-454
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