Linear Algebra, Geometry and Transformation

The Essentials of a First Linear Algebra Course and More

Linear Algebra, Geometry and Transformation provides students with a solid geometric grasp of linear transformations. It stresses the linear case of the inverse function and rank theorems and gives a careful geometric treatment of the spectral theorem.

An Engaging Treatment of the Interplay among Algebra, Geometry, and Mappings

The text starts with basic questions about images and pre-images of mappings, injectivity, surjectivity, and distortion. In the process of answering these questions in the linear setting, the book covers all the standard topics for a first course on linear algebra, including linear systems, vector geometry, matrix algebra, subspaces, independence, dimension, orthogonality, eigenvectors, and diagonalization.

A Smooth Transition to the Conceptual Realm of Higher Mathematics

This book guides students on a journey from computational mathematics to conceptual reasoning. It takes them from simple “identity verification” proofs to constructive and contrapositive arguments. It will prepare them for future studies in algebra, multivariable calculus, and the fields that use them.

Table of Contents

CHAPTER 1 – Vectors, Mappings, and Linearity
CHAPTER 2 – Solving Linear Systems
CHAPTER 3 – Linear Geometry
CHAPTER 4 – The Algebra of Matrices
CHAPTER 5 – Subspaces
CHAPTER 6 – Orthogonality
CHAPTER 7 – Linear Transformation
APPENDIX A – Determinants
APPENDIX B – Proof of the Spectral Theorem
APPENDIX C – Lexicon