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A Linear Algebra Problem Book

Paul R. Halmos
Mathematical Association of America
Publication Date: 
Number of Pages: 
Dolciani Mathematical Expositions 16
BLL Rating: 

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on

This is more than a problem book: it is a complete inquiry-based course in linear algebra. It operates by looking at specific examples (usually matrices, sometimes vector spaces) to derive some conjectures and generalizations and then asking the reader to prove or disprove these. The problems are quite hard! Even if you know the subject well you may have to think a bit about some of them.

The book is much more chatty than the usual problem book. It’s full of statements like “Try this…” or “That doesn’t work because…”. The structure is the same as Halmos’s A Hilbert Space Problem Book, with a long problem section, a short hint section, and a long solution section. The narrative weaves back and forth between the problem section and the solution section, so you should read the solutions even if you solve the problems by yourself.

The book is to a large extent a re-working of Halmos’s earlier book Finite-Dimensional Vector Spaces, covering the same topics, but rearranged as a linked series of problems. There are no numerical exercises, but usually numerical examples are used to help discover the theorems. I find it much more accessible than the earlier book, both because of the extra narrative and because it is driven by examples.

One weakness of this book is that it is very insular: it never tells us where linear algebra came from or where it can go. The various properties defined and explored here were historically driven by problems in other areas of mathematics, in particular Hilbert spaces and operator theory, but we are not shown that. Rather than tying the subject to outside problems, the book uses a “characterization” approach: it defines a series of properties, and for each one attempts to characterize the objects that have that property and relate the properties to other properties that have been defined.

A Very Bad Feature of the book is that it has no index. If, for example, you read in the hint to problem 148 that you should use the spectral theorem but don’t know what that is, you may have a hard time finding out.

Is this a complete textbook, or just a handy supplement? That depends on your purpose. For most courses today it would be inadequate by itself because it doesn’t have any applications or any numerical techniques. If your course is really Matrix Algebra and not Linear Algebra, as many are, this is not the book for you! But it does cover everything that is strictly within the theory of linear algebra, in an interesting and fun way, and if you have decided you are going to focus on proofs and problem-solving, then this book has everything you need.

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.

  • Preface
  • Chapter 1. Scalars
    1. Double addition
    2. Half double addition
    3. Exponentiation
    4. Complex numbers
    5. Affine transformations
    6. Matrix multiplication
    7. Modular multiplication
    8. Small operations
    9. Identity elements
    10. Complex inverses
    11. Affine inverses
    12. Matrix inverses
    13. Abelian groups
    14. Groups
    15. Independent group axioms
    16. Fields
    17. Addition and multiplication in fields
    18. Distributive failure
    19. Finite fields
  • Chapter 2. Vectors
    1. Vector spaces
    2. Examples
    3. Linear combinations
    4. Subspaces
    5. Unions of subspaces
    6. Spans
    7. Equalities of spans
    8. Some special spans
    9. Sums of subspaces
    10. Distributive subspaces
    11. Total sets
    12. Dependence
    13. Independence
  • Chapter 3. Bases
    1. Exchanging bases
    2. Simultaneous complements
    3. Examples of independence
    4. Independence over R and Q
    5. Independence in C2
    6. Vectors common to different bases
    7. Bases in C3
    8. Maximal independent sets
    9. Complex as real
    10. Subspaces of full dimension
    11. Extended bases
    12. Finite-dimensional subspaces
    13. Minimal total sets
    14. Existence of minimal total sets
    15. Infinitely total sets
    16. Relatively independent sets
    17. Number of bases in a finite vector space
    18. Direct sums
    19. Quotient spaces
    20. Dimension of a quotient space
    21. Additivity of dimension
  • Chapter 4. Transformations
    1. Linear transformation
    2. Domain and range
    3. Kernel
    4. Composition
    5. Range inclusion and factorization
    6. Transformations as vectors
    7. Invertibility
    8. Invertibility examples
    9. Determinants: 2 x 2
    10. Determinants: n x n
    11. Zero-one matrices
    12. Invertible matrix bases
    13. Finite-dimensional invertibility
    14. Matrices
    15. Diagonal matrices
    16. Universal commutativity
    17. Invariance
    18. Invariant complements
    19. Projections
    20. Sums of projections
    21. Not quite idempotence
  • Chapter 5. Duality
    1. Linear functionals
    2. Dual spaces
    3. Solution of equations
    4. Reflexivity
    5. Annihilators
    6. Double annihilators
    7. Adjoints
    8. Adjoints of projections
    9. Matrices of adjoints
  • Chapter 6. Similarity
    1. Change of basis: vectors
    2. Change of basis: coordinates
    3. Similarity: transformations
    4. Similarity: matrices
    5. Inherited similarity
    6. Similarity: real and complex
    7. Rank and nullity
    8. Similarity and rank
    9. Similarity of transposes
    10. Ranks of sums
    11. Ranks of products
    12. Nullities of sums and products
    13. Some similarities
    14. Equivalence
    15. Rank and equivalence
  • Chapter 7. Canonical Forms
    1. Eigenvalues
    2. Sums and products of eigenvalues
    3. Eigenvalues of products
    4. Polynomials in eigenvalues
    5. Diagonalizing permutations
    6. Polynomials in eigenvalues converse
    7. Multiplicities
    8. Distinct eigenvalues
    9. Comparison of multiplicities
    10. Triangularization
    11. Complexification
    12. Unipotent transformation
    13. Nilpotence
    14. Nilpotent products
    15. Nilpotent direct sums
    16. Jordan form
    17. Minimal polynomials
    18. Noncommutative Lagrange interpolation
  • Chapter 8. Inner Product Spaces
    1. Inner products
    2. Polarization
    3. The Pythagorean theorem
    4. The parallelogram law
    5. Complete orthonormal sets
    6. Schwarz inequality
    7. Orthogonal complements
    8. More linear functional
    9. Adjoints on inner product spaces
    10. Quadratic forms
    11. Vanishing quadratic forms
    12. Hermitian transformations
    13. Skew transformations
    14. Real Hermitian forms
    15. Positive transformations
    16. Positive inverses
    17. Perpendicular projections
    18. Projections on C x C
    19. Projection order
    20. Orthogonal projections
    21. Hermitian eigenvalues
    22. Distinct eigenvalues
  • Chapter 9. Normality
    1. Unitary transformations
    2. Unitary matrices
    3. Unitary involutions
    4. Unitary triangles
    5. Hermitian diagonalization
    6. Square roots
    7. Polar decomposition
    8. Normal transformations
    9. Normal diagonalizability
    10. Normal commutativity
    11. Adjoint commutativity
    12. Adjoint intertwining
    13. Normal products
    14. Functions of transformations
    15. Gramians
    16. Monotone functions
    17. Reducing ranges and kernels
    18. Truncated shifts
    19. Non-positive square roots
    20. Similar normal transformations
    21. Unitary equivalence of transposes
    22. Unitary and orthogonal equivalence
    23. Null convergent powers
    24. Power boundedness
    25. Reduction and index 2
    26. Nilpotence and reduction
  • Hints
  • Solutions