Elementary Linear Algebra

Citation

Anton, H., & Kaul, A. (2019). Elementary Linear Algebra (12th ed.). John Wiley & Sons.

Chapter Summary

Chapter 1: Systems of Linear Equations and Matrices

The longest chapter starts from linear systems and Gaussian elimination, then develops matrix operations, inverses and their algebraic properties, elementary matrices, diagonal, triangular and symmetric matrices, and an introduction to linear transformations and their compositions. It ends with two application sections: applications of linear systems (network analysis, electrical circuits, balancing chemical equations, polynomial interpolation) and Leontief input-output models.

Chapter 2: Determinants

Determinants are introduced by cofactor expansion, then evaluated by row reduction, and their properties are developed together with Cramer's rule. The chapter has only three sections.

Chapter 3: Euclidean Vector Spaces

This chapter moves from vectors in 2-space, 3-space and n-space to norm, dot product and distance in R^n, orthogonality, the geometry of linear systems and the cross product.

Chapter 4: General Vector Spaces

The ideas of Chapter 3 are generalised to abstract real vector spaces: subspaces, spanning sets, linear independence, coordinates and basis, dimension and change of basis. The chapter closes with the row space, column space and null space, and with rank, nullity and the fundamental matrix spaces.

Chapter 5: Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors, diagonalization and complex vector spaces are followed by two applications: differential equations, and dynamical systems and Markov chains.

Chapter 6: Inner Product Spaces

The chapter treats inner products, angle and orthogonality in inner product spaces, the Gram–Schmidt process and QR-decomposition, best approximation and least squares, mathematical modelling using least squares, and function approximation and Fourier series.

Chapter 7: Diagonalization and Quadratic Forms

Orthogonal matrices and orthogonal diagonalization lead to quadratic forms and to optimization using quadratic forms. A final section covers Hermitian, unitary and normal matrices.

Chapter 8: General Linear Transformations

Linear transformations between general vector spaces are treated here, with compositions and inverse transformations, isomorphism, matrices for general linear transformations, similarity, and the geometry of matrix operators.

Chapter 9: Numerical Methods

LU-decompositions and the power method are followed by a comparison of procedures for solving linear systems, the singular value decomposition, and data compression using the singular value decomposition.

Supplemental topics and appendices

The contents list online supplemental topics (three on linear programming, and others on vectors in plane geometry, equilibrium of rigid bodies, the assignment problem, the determinant function and Leontief economic models). Appendix A covers working with proofs, and Appendix B complex numbers. Answers to exercises and an index end the volume.

Key Concepts

Chapters 1–2: Systems, Matrices and Determinants

• Gaussian elimination: the systematic reduction of a linear system to find its solutions.
• Matrix inverse and elementary matrices: the link between row operations and invertibility.
• Cofactor expansion and Cramer's rule: determinant tools for small systems and for testing invertibility.

Chapters 3–4: Vector Spaces

• Dot product, norm and orthogonality: the geometry of R^n.
• Linear independence, basis and dimension: the structure of a general vector space.
• Row space, column space and null space; rank and nullity: the fundamental matrix spaces.

Chapters 5–7: Eigenvalues, Inner Products and Diagonalization

• Eigenvalue and eigenvector: the directions a matrix stretches without turning.
• Gram–Schmidt process and least squares: building orthogonal bases and fitting data.
• Orthogonal diagonalization and quadratic forms: diagonalization of symmetric matrices and its use in optimization.

Chapters 8–9: Transformations and Numerical Methods

• Similarity and matrices of general linear transformations: representing a transformation relative to different bases.
• LU-decomposition, power method and singular value decomposition: computational factorizations and their uses.

Critical Analysis

Strengths

• Pedagogy first. Wiley describes the book as an elementary treatment suitable for a first undergraduate course, with the aim of presenting the fundamentals in the clearest possible way.
• Calculus is optional. Calculus is not a prerequisite; exercises and examples that use it are clearly labelled and can be omitted without loss of continuity.
• Rich exercise sets. In the Chapter 1 sample, each section ends with an exercise set that includes true-false exercises, and most add a "Working with Proofs" group and a "Working with Technology" group, so computation, reasoning and software use are all practised.
• Geometry before abstraction. Euclidean spaces (Chapter 3) come before general vector spaces (Chapter 4), and determinants come early (Chapter 2).
• Applications throughout. Chapter 1 ends with network analysis, circuits, chemical equations, polynomial interpolation and Leontief models. Later chapters apply the theory to differential equations, Markov chains, least squares, Fourier series, quadratic forms and data compression.
• Proof support. Appendix A, "Working with Proofs", helps students begin reading and writing proofs.

Limitations

• Abstraction comes late. General linear transformations do not appear until Chapter 8, after eigenvalues, so students who want a proof-first treatment organised around linear maps (as in Axler) will find the order different.
• Numerical methods are brief. Chapter 9 has five sections and roughly 40 pages, enough to meet the main factorizations but not a course in numerical linear algebra.
• Some material is online. The supplemental topics, including the three on linear programming, are not in the printed text.
• Size. The book runs to over 500 pages, and most courses will not use all of it.

Where It Fits in the Course

Anton is the conventional applications-oriented first course in the linear algebra list. Strang's Introduction to Linear Algebra builds the subject around the four fundamental subspaces and leads to the singular value decomposition; Hefferon's Linear Algebra is a free text for a first course that proves its results while building mathematical maturity. Anton's order (systems first, then geometry, then abstraction) suits students who want to compute fluently before they see axioms.

For more theory, Axler's Linear Algebra Done Right develops the subject through vector spaces and linear maps and postpones determinants; Anthony and Harvey's Linear Algebra: Concepts and Methods is an undergraduate text that, like Anton, moves from matrices and systems to vector spaces, diagonalisation and orthogonality. Horn and Johnson's Matrix Analysis is a graduate reference far beyond Anton's scope.

How to Use This Book

The book is for first-year and second-year undergraduates in mathematics, science, engineering and economics. The prerequisite is school algebra and geometry; calculus is helpful only for the marked exercises.

Chapters 1 to 4 form the core of a first course, followed by Chapter 5 (eigenvalues) and Chapter 6 (inner product spaces). Chapter 7 gives the results on symmetric matrices and quadratic forms. Chapters 8 and 9 and the later sections of Chapters 1 and 5 can be taken as options, depending on whether the course emphasises theory (Chapter 8) or computation and applications (Chapter 9, Sections 1.10 and 1.11, 5.4 and 5.5).

Work the exercises in the order given, and do at least a few of the true-false and "Working with Proofs" questions in each set: they are a good check on whether the ideas are understood. Use the answers at the back to check your work, and the technology exercises to build fluency with software.