1st Edition

Linear Algebra, Geometry and Transformation

By Bruce Solomon Copyright 2015
    474 Pages 63 B/W Illustrations
    by Chapman & Hall

    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.

    Print Versions of this book also include access to the ebook version.


    Vectors, Mappings and Linearity
    Numeric Vectors
    Mappings and Transformations
    The Matrix of a Linear Transformation

    Solving Linear Systems
    The Linear System
    The Augmented Matrix and RRE Form
    Homogeneous Systems in RRE Form
    Inhomogeneous Systems in RRE Form
    The Gauss-Jordan Algorithm
    Two Mapping Answers

    Linear Geometry
    Geometric Vectors
    Geometric/Numeric Duality
    Dot-Product Geometry
    Lines, Planes, and Hyperplanes
    System Geometry and Row/Column Duality

    The Algebra of Matrices
    Matrix Operations
    Special Matrices
    Matrix Inversion
    A Logical Digression
    The Logic of the Inversion Algorithm

    Basic Examples and Definitions
    Spans and Perps
    Perp/Span Conversion
    Dimension and Rank

    Four Subspaces, 16 Questions
    Orthonormal Bases
    The Gram-Schmidt Algorithm

    Linear Transformation
    Kernel and Image
    The Linear Rank Theorem
    Eigenvalues and Eigenspaces: Calculation
    Eigenvalues and Eigenspaces: Similarity
    Diagonalizability and the Spectral Theorem
    Singular Value Decomposition

    Appendix A: Determinants
    Appendix B: Proof of the Spectral Theorem
    Appendix C: Lexicon



    Bruce Solomon is a professor in the Department of Mathematics at Indiana University Bloomington, where he often teaches linear algebra. He has held visiting positions at Stanford University and in Australia, France, and Israel. His research articles explore differential geometry and geometric variational problems. He earned a PhD from Princeton University.

    "All the standard topics of a first course are covered, but the treatment omits abstract vector spaces. … What is unusual is the author's aim to interpret every concept and result geometrically, thus motivating the student to learn to visualize what is going on, rather than just relying on calculations. This is a strong and useful feature. … The book has very many practice sections with over 500 exercises, most of them numerical. … As the author mentions in the preface, it was his aim to provide a sound mathematical introduction, and in the reviewer's opinion he has succeeded in doing this."
    Zentralblatt MATH 1314