A Portrait of Linear Algebra


ISBN: 978-1-4652-3267-0
ISBN: 978-1-4652-2210-7

Written in a light, friendly, and conversational tone, A Portrait of Linear Algebra contains material for a standard introductory course in Linear Algebra, as well as many topics covered in a more advanced treatment of the subject. Almost all Theorems are either proven or left as an exercise for the reader. Available in both print and eBook formats, A Portrait of Linear Algebra includes a summary of pertinent definitions and theorems at the end of each section, and a chapter summary provides the student with a big picture of each major topic. In addition, the book provides a wide variety of exercises, including both computational and theoretical problems.

What makes A Portrait of Linear Algebra different?

  • An Emphasis on Proofs: For students in many colleges, Linear Algebra is their first encounter with rigorous mathematical proofs. This book does not assume that the student will miraculously get the hang of writing a proof all on their own. Instead, the tone is set in Chapter Zero, where we guide the student through the various basic proof techniques. We use the field axioms of the real number system as the main model, with several examples that the student can emulate in their own proofs. These are later generalized in the context of linear algebra in Chapters 1 and 2. We emphasize that students should learn definitions, good notation, and the use of appropriate words and phrases in writing a proof throughout the book. More difficult proof exercises are broken down into small steps for the student to handle more easily. This axiomatic treatment of Linear Algebra is continued into Chapter 3, where we study abstract vector spaces and their linear transformations, and in Chapter 7, where we generalize the dot product into inner products on abstract spaces.
  • A Geometric Component: Vectors and the effect of linear transformations are described visually as well as
  • A Comprehensive Discussion of Complex Euclidean Spaces: Chapter 8 develops the theory of linear transformations on complex spaces, including a discussion of Hermitian, skew-Hermitian, and unitary matrices, a complete proof of Schur’s Lemma and The Spectral Theorem for Normal Matrices.
  • A Complete Development of the Fundamental Theorem of Linear Algebra and the Singular Value Decomposition: The book culminates with a discussion of various applications of Linear Algebra, the theory of Quadratic Forms, and a complete proof of The Fundamental Theorem of Linear Algebra and its computational twin, The Singular Value Decomposition, and its applications.