The volume under review brings together 27 classic articles on linear algebra and some of its relevant applications to computational science, applied mathematics, and calculus, each introduced by a new essay. Some of the central themes of this book are: the four fundamental subspaces and perfect bases; an effective way to introduce the exponential function; the finite element method and its accuracy; wavelets and signal processing; the fundamental theorem of calculus; the joint spectral radius; and a great variety of matrices (Toeplitz, circulant, banded, incidence, graph Laplacian, and Pascal matrices). The present volume contains the following parts. Part I. Essays in linear algebra: 1. Perfect bases for the four subspaces; 2. Four subspaces from a graph; 3. My favourite matrix; 4. The ultimate in echelon forms; 5. Diagonalizing a symmetric matrix; 6. Pascal’s triangle in three matrices; 7. Can the factors be sparse?; 8. The eigenvalues of $AB + BA$; 9. Factoring into very special matrices; 10. Toeplitz matrices and circulants; 11. The joint spectral radius of A and B; 12. Banded matrices have remarkable inverses. Part II. Essays in calculus: 13. The fundamental theorem of calculus; 14. The great function of calculus; 15. Thin strips and small triangles in calculus; 16. Chaos in Newton’s method; 17. Optical illusions for $y = \sin n$. Part III. Essays in applied mathematics: 18. The principles of applied mathematics; 19. A minimum equals a maximum; 20. Aiming directly at the solution; 21. A chair through a door. Part IV. Essays in computational science: 22. Eight cosine transforms; 23. The ideas behind finite elements; 24. The Strang-Fix conditions; 25. Wavelets in music and television; 26. Scaling functions lead to wavelets; 27. Old and new splitting methods; An essay about books. All essays present historical background, the current state, and unsolved problems. In summary, this delightful book is a most welcome addition to the library of any mathematician with an interest in applied linear analysis.

Reviewer:

Vicenţiu D. Rădulescu (Craiova)