A Portrait of Linear Algebra

A Portrait of Linear Algebra uses a unified approach that gives the reader an integrated view of topics in this subject. It covers all the material in a standard introductory course on linear algebra, with enough material for two full semesters. 

Unique Features: 

• Emphasizes the importance of learning how to read and write proofs.
• Key definitions and theorems found in the discussions are restated in the section and chapter summaries for easy reference.
• Linear algebra is developed by generalizing the field axioms for the real numbers to Euclidean spaces.
• A vector-centered approach is used throughout the book.
• Fundamental concepts such as spanning, linear independence, basis, dimension, one-to-one and onto linear transformations and isomorphisms are discussed in the first two chapters.
• Permutation Theory is developed and applied to the construction of determinants.
• Covers the eigentheory of matrices, as well as linear transformations of finite-dimensional vector spaces, along with the related concept of similarity.
• Several key theorems which are not usually found in introductory books are thoroughly developed and proved: The Isomorphism Theorems of Emy Noether, Schur’s Lemma, The Spectral Theorem for Normal Matrices, The Fundamental Theorem of Linear Algebra, and The Singular Value Decomposition.
• Creative and unusual examples and exercises are found throughout the book.
• The exercises include a wide range of computational problems, as well as proofs of theorems.
• Difficult proof exercises are outlined for the student, and hints are provided.
• The book is written in a reader-friendly conversational style, ideal for independent study.

An optional, introductory chapter discusses common logical techniques that can be used to prove theorems in linear algebra, such as, how to apply axioms and definitions to prove basic properties of any mathematical structure, case-by-case analyses, proof by contradiction, proof by contrapositive, and mathematical induction. This is of particular importance to students who have never had a course on proofs before.

The text discusses existential and universal quantifies and how to write the inverse, converse and contrapositive of an implication. Examples are motivated by an axiomatic development of the real number system. Most are analogous to similar statements later seen in the context of vectors. A discussion of the complete set of axioms of the real number system is also presented in Appendix A.

Linear Algebra is defined as the study of vector spaces, their structure, and the linear transformations that map one vector space to another. Euclidean spaces and their linear transformations are introduced in the first two main chapters. This easily allows all constructions to be rewritten for general vectors spaces in the third chapter, the main examples being matrix spaces and function spaces (polynomials, continuous functions and differentiable functions).

To explore the subspace structure of vector spaces, cosets and quotient spaces of vector spaces are constructed. How to construct a basis for the join and intersection of two subspaces given a basis for each subspace, as well as how to construct a basis for the image or preimage of a subspace under a linear transformation is also shown. Some of these constructions are rarely seen in introductory textbooks, but they will allow students to explore the consequences of the three Isomorphism Theorems.

Complex Euclidean spaces, linear transformations, matrices and inner products are fully discussed, with the goal of studying Hermitian, skew-Hermitian, unitary and normal matrices. Schur’s Lemma and the Spectral Theorems are proven completely.

The capstone of the book is the development of the theory of positive-definite and semi-definite matrices, which allows students to prove the Fundamental Theorem of Linear Algebra and The Singular Value Decomposition, or SVD. The Fundamental Theorem neatly wraps up several key ideas in one beautiful package: rowspace, columnspace and nullspace, eigenspaces and orthogonality. The SVD is one of the most important workhorses in modern computing and data compression.