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Introduction to Linear Algebra

Gilbert Strang
Wellesley-Cambridge Press
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
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This is an erudite and discursive introduction to linear algebra, weighted heavily toward matrices and systems of linear equations. The author has an expansive view of linear algebra, and from time to time draws in some calculus, Fourier series, wavelets, and function spaces, but the approach is always very concrete. The book doesn't skimp on the techniques of linear algebra, and there are seemingly endless examples of LU-decomposition and other numeric work, as well as a fairly extensive Chapter 9 on numerical methods. But the book also does a good job of moving up and down between various levels of abstraction, according to which level makes the problem at hand easier to comprehend, and geometrical examples and rotations play an important role in the exposition.

This book is the text for Massachusetts Institute of Technology’s Linear Algebra course 18.06, whose goals are “using matrices and also understanding them.” There’s enough material in the book for a year-long course, and the MIT course covers primarily the first seven chapters. There is a great deal of free supplemental material available on the MIT web site for the course, including videos of Strang’s lectures in the course, interactive computer demonstrations, and past exams and problem sets. The present book is complete in itself, but many students will appreciate the additional resources.

Each section of the book is structured as several pages of narrative, usually including motivational examples, followed by a several-point summary, followed by more worked examples and a problem set that focuses on drill but also includes open-ended questions suggested by the examples, and closing with a collection of “challenge problems” that really are challenging. Proofs of nearly all results are included, although the book does not use a “theorem–proof” format and doesn’t make a big deal of proving things. One potential drawback of the book’s discursive nature is that it may be hard to use for reference and review: a topic may be revisited several times to look at different aspects, so the material on a particular topic may not be wrapped up in a tidy package.

Chapter 6, on eigenvalues and eigenvectors, is especially strong and is the centerpiece of the book. It not only teaches you to think about what eigenvectors really mean geometrically and in applications, but pulls together the earlier material on spaces and linear transformations, showing why it is important sometimes to think at a higher level of abstraction.

Very Good Feature: The author frequently points out which of the concepts are the most important ones, so you don’t have to wonder whether (for example) unitary matrices are something you really care about. Some sample quotes: “The Singular Value Decomposition is a highlight of linear algebra.” (p. 363) “It is no exaggeration to say that these [symmetric matrices] are the most important matrices the world will ever see.” (p. 330)

Compared to Strang’s other linear algebra book, Linear Algebra and Its Applications, both books cover roughly the same material, but the present book is aimed lower, with more drill exercises and more handholding in the form of more-detailed explanations and more worked examples. The Applications book has more applications, as you would expect, and they are integrated into the narrative where they naturally arise rather than being collected in a separate chapter as in the present book.

Bottom line: an intriguing and challenging text that does a good job of pulling together the disparate pieces of linear algebra, but that may overemphasize matrices at the expense of spaces and linear transformations.

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.

  • 1 Introduction to Vectors
    • 1.1 Vectors and Linear Combinations
    • 1.2 Lengths and Dot Products
    • 1.3 Matrices
  • 2 Solving Linear Equations
    • 2.1 Vectors and Linear Equations
    • 2.2 The Idea of Elimination
    • 2.3 Elimination Using Matrices
    • 2.4 Rules for Matrix Operations
    • 2.5 Inverse Matrices
    • 2.6 Elimination = Factorization: A = LU
    • 2.7 Transposes and Permutations
  • 3 Vector Spaces and Subspaces
    • 3.1 Spaces of Vectors
    • 3.2 The Nullspace of A: Solving Ax = 0
    • 3.3 The Rank and the Row Reduced Form
    • 3.4 The Complete Solution to Ax = b
    • 3.5 Independence, Basis and Dimension
    • 3.6 Dimensions of the Four Subspaces
  • 4 Orthogonality
    • 4.1 Orthogonality of the Four Subspaces
    • 4.2 Projections
    • 4.3 Least Squares Approximations
    • 4.4 Orthogonal Bases and Gram-Schmidt
  • 5 Determinants
    • 5.1 The Properties of Determinants
    • 5.2 Permutations and Cofactors
    • 5.3 Cramer’s Rule, Inverses, and Volumes
  • 6 Eigenvalues and Eigenvectors
    • 6.1 Introduction to Eigenvalues
    • 6.2 Diagonalizing a Matrix
    • 6.3 Applications to Differential Equations
    • 6.4 Symmetric Matrices
    • 6.5 Positive Definite Matrices
    • 6.6 Similar Matrices
    • 6.7 Singular Value Decomposition (SVD)
  • 7 Linear Transformations
    • 7.1 The Idea of a Linear Transformation
    • 7.2 The Matrix of a Linear Transformation
    • 7.3 Diagonalization and the Pseudoinverse
  • 8 Applications
    • 8.1 Matrices in Engineering
    • 8.2 Graphs and Networks
    • 8.3 Markov Matrices, Population, and Economics
    • 8.4 Linear Programming
    • 8.5 Fourier Series: Linear Algebra for Functions
    • 8.6 Linear Algebra for Statistics and Probability
    • 8.7 Computer Graphics
  • 9 Numerical Linear Algebra
    • 9.1 Gaussian Elimination in Practice
    • 9.2 Norms and Condition Numbers
    • 9.3 Iterative Methods and Preconditioners
  • 10 Complex Vectors and Matrices
    • 10.1 Complex Numbers
    • 10.2 Hermitian and Unitary Matrices
    • 10.3 The Fast Fourier Transform
  • Solutions to Selected Exercises
  • Conceptual Questions for Review
  • Glossary: A Dictionary for Linear Algebra
  • Matrix Factorizations
  • Index
  • Teaching Codes
  • Linear Algebra in a Nutshell