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Linear Algebra for Everyone

Lorenzo Robbiano
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
, on

This is a conventional introduction to matrices that the author has attempted to make more accessible by garnishing it with palindromes, jokes, and aphorisms. It is a translation of the 2007 Italian-language work Algebra Lineare per tutti (also published by Springer). The translation is smooth and easy to read, although many of the jokes are left untranslated.

From the title one might guess that this is a text for a hypothetical Linear Algebra Appreciation course, but the author admits on p. x that the focus is much narrower. The motivation for the book was to produce a unified text that all departments would use, rather than have each teach its own flavor of linear algebra. I think this unification is unsuccessful; the applications are very generic and other departments would consider it slanted too much toward pure mathematics and not having anything special that the departmental students need to know.

The book takes a very concrete approach. As motivation, it starts out with a selection of practical problems that can be expressed in terms of systems of linear equations. The treatment is slanted very much toward matrices and numerical work and avoids the linear-spaces interpretation, except in Chapter 4 where the book explores geometric vectors and rotations and projections.

Assuming that the work is in fact aimed at linear algebra students and not at “everyone”, its big weakness is that it doesn’t have enough exercises and worked examples. This omission would make it unsuitable for most American college-level courses. Other popular textbooks such as Strang’s Introduction to Linear Algebra or Lay’s Linear Algebra and Its Applications would be better choices for a college course. They are only slightly more expensive than the present book, have many applications from all areas, and have an adequate number of exercises and worked examples.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.

  • Dedication
  • Foreword
  • Introduction
  • Numerical and Symbolic Computations
    • The equation ax = b. Let’s try to solve it
    • The equation ax = b. Be careful of mistakes
    • The equation ax = b. Let’s manipulate the symbols
  • Exercises
  • Part I
  1. Systems of Linear Equations and Matrices
    • 1.1 Examples of Systems of Linear Equations
    • 1.2 Vectors and Matrices
    • 1.3 Generic Systems of Linear Equations and Associated Matrices
    • 1.4 The Formalism of Ax = b
    • Exercises
  2. Operations with Matrices
    • 2.1 Sum and the product by a number
    • 2.2 Row by column product
    • 2.3 How much does it cost to multiply two matrices?
    • 2.4 Some properties of the product of matrices
    • 2.5 Inverse of a matrix
    • Exercises
  3. Solutions of Systems of Linear Equations
    • 3.1 Elementary Matrices
    • 3.2 Square Linear Systems, Gaussian Elimination
    • 3.3 Effective Calculation of Matrix Inverses
    • 3.4 How much does Gaussian Elimination cost?
    • 3.5 The LU Decomposition
    • 3.6 Gaussian Elimination for General Systems of Linear Equations
    • 3.7 Determinants
    • Exercises
  4. Coordinate Systems
    • 4.1 Scalars and Vectors
    • 4.2 Cartesian Coordinates
    • 4.3 The Parallelogram Rule
    • 4.4 Orthogonal Systems, Areas, Determinants
    • 4.5 Angles, Moduli, Scalar Products
    • 4.6 Scalar Products and Determinants in General
    • 4.7 Change of Coordinates
    • 4.8 Vector Spaces and Bases
    • Exercises
  • Part II
  1. Quadratic Forms
    • 5.1 Equations of the Second Degree
    • 5.2 Elementary Operations on Symmetric Matrices
    • 5.3 Quadratic Forms, Functions, Positivity
    • 5.4 Cholesky Decomposition
    • Exercises
  2. Orthogonality and Orthonormality
    • 6.1 Orthonormal Tuples and Orthonormal Matrices
    • 6.2 Rotations
    • 6.3 Subspaces, Linear Independence, Rank, Dimension
    • 6.4 Orthonormal Bases and the Gram-Schmidt Procedure
    • 6.5 The QR Decomposition
    • Exercises
  3. Projections, Pseudoinverses and Least Squares
    • 7.1 Matrices and Linear Transformations
    • 7.2 Projections
    • 7.3 Least Squares and Pseudoinverses
    • Exercises
  4. Endomorphisms and Diagonalization
    • 8.1 An Example of a Plane Linear Transformation
    • 8.2 Eigenvalues, Eigenvectors, Eigenspaces and Similarity
    • 8.3 Powers of Matrices
    • 8.4 The Rabbits of Fibonacci
    • 8.5 Differential Systems
    • 8.6 Diagonalizability of Real Symmetric Matrices
    • Exercises
  • Part III
  • Appendix
  • Conclusion?
  • References
  • Index