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Linear Transformations on Vector Spaces

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Scott Kaschner, Butler University

Amber Russell, Butler University

Copyright Year: 2023

Publisher: PALNI

Language: English

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Table of Contents

  • Preface
  • Chapter 0: Functions on Sets
  • Chapter 1: Vector Spaces
  • Chapter 2: Bases
  • Chapter 3: Linear Transformation
  • Chapter 4: More Fun with Matrices
  • Chapter 5: Square Matrices and Invariant Subspaces
  • Appendix
  • Glossary

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About the Book

Linear Transformations on Vector Spaces serves primarily as a textbook for undergraduate Linear Algebra courses. While standard Linear Algebra books begin by focusing on solving systems of linear equations and associated procedural skills, our book begins by developing a conceptual framework for the topic using the central objects, vector spaces and linear transformations. It covers the same concepts, skills, and, applications as conventional texts in a one-semester course, but students walk away with a much richer and more useful mastery of the topics. The book is structured to facilitate the implementation of the flipped classroom. The text features a continuous narrative to illuminate the big picture of the material and is written to help students develop their textbook reading skills. Also, there are “Explorations” scattered throughout each section; these are quick first examples intended for students to complete while reading before class meetings. Additional materials include section overview homework assignments and worksheets that can be used for in-class practice.

About the Contributors


Scott Kaschner is an Associate Professor and the Department Chair in the Department of Mathematical Sciences at Butler University.  He earned a Ph.D. in Mathematics from Purdue University (at Indiana University Purdue University Indianapolis) and did a postdoc at the University of Arizona.  His primary research area is complex dynamical systems in one or several variables; he also does work in operator theory, undergraduate mathematics education, professional development for mathematics educators, and mathematical biology.

Amber Russell is an Associate Professor in the Mathematical Sciences Department at Butler University and the current Director of Butler’s Engineering Dual Degree Program.  She received her Ph.D. in Mathematics from Louisiana State University.  Her research area is geometric representation theory, but she particularly enjoys studying the combinatorics associated to Springer theory.  She was a postdoctoral researcher for the Algebra Research Group at the University of Georgia, where she first taught Linear Algebra, both an applied and a theoretical version.  She is responsible for all unicorn images in this text, including the really unfortunate original one in Chapter 2.

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