Linear Algebra: Concepts and Methods

Citation

Anthony, M., & Harvey, M. (2012). Linear Algebra: Concepts and Methods. Cambridge University Press.

Chapter Summary

The book is a first-year university course in linear algebra, written for students from any discipline, with some later material added. Each chapter follows one pattern: the main text is interspersed with short "activities" that ask the reader to pause and check an idea, and ends with a list of learning outcomes, comments on the activities, worked exercises and further problems. A short revision chapter precedes Chapter 1.

Preliminaries: Before We Begin

A brief revision of sets and set notation, numbers, mathematical terminology, basic algebra, trigonometry and a little logic. The authors expect most of it to be familiar and suggest looking up anything that is not in a basic mathematics text.

Chapter 1: Matrices and Vectors

Matrices come first: addition, scalar multiplication, multiplication, matrix algebra, inverses, powers, and the transpose and symmetric matrices. The chapter then turns to vectors in Rⁿ, with a section on developing geometric insight and treatments of lines, planes in R³ and lines and hyperplanes in Rⁿ.

Chapter 2: Systems of Linear Equations

Linear systems are solved by row operations and Gaussian elimination. The chapter ends with homogeneous systems and the null space.

Chapter 3: Matrix Inversion and Determinants

The inverse is computed first by row operations and later by cofactors, with determinants and their properties in between. A final section applies the machinery to Leontief input–output analysis.

Chapter 4: Rank, Range and Linear Equations

The rank of a matrix is linked to the solvability of linear systems, and the range is introduced.

Chapter 5: Vector Spaces

The abstract setting arrives here, with vector spaces, subspaces and linear span, after the matrix and Rⁿ material that motivates it.

Chapter 6: Linear Independence, Bases and Dimension

Linear independence, bases, coordinates and dimension are developed, with a closing section on basis and dimension in Rⁿ.

Chapter 7: Linear Transformations and Change of Basis

Linear transformations, their range and null space, coordinate change, and change of basis and similarity connect the abstract theory back to matrices.

Chapter 8: Diagonalisation

Eigenvalues and eigenvectors lead to the diagonalisation of a square matrix and the question of when diagonalisation is possible.

Chapter 9: Applications of Diagonalisation

Diagonalisation is used for powers of matrices, systems of difference equations and linear systems of differential equations.

Chapter 10: Inner Products and Orthogonality

Inner products, orthogonality, orthogonal matrices and the Gram–Schmidt orthonormalisation process.

Chapter 11: Orthogonal Diagonalisation and Its Applications

Symmetric matrices are orthogonally diagonalised, which leads to a long section on quadratic forms.

Chapter 12: Direct Sums and Projections

Direct sums of two subspaces, orthogonal complements, projections and orthogonal projections, projection onto the range of a matrix, minimising the distance to a subspace and fitting functions to data by least squares.

Chapter 13: Complex Matrices and Vector Spaces

Complex numbers, complex vector spaces and matrices, complex inner product spaces, Hermitian conjugates, unitary diagonalisation and normal matrices, and spectral decomposition.

Key Concepts

Chapters 1–4: Matrices, Equations and Rank

• Gaussian elimination: reducing a system by row operations to read off its solutions.
• Null space and range: the solutions of Ax = 0 and the set of vectors Ax, which together describe when a system is solvable and how many solutions it has.
• Determinant and cofactors: a number attached to a square matrix, used to test invertibility and to compute inverses.

Chapters 5–7: Vector Spaces and Linear Maps

• Basis and dimension: a linearly independent spanning set, whose size is the dimension of the space.
• Change of basis and similarity: how a linear transformation's matrix changes when the basis changes.

Chapters 8–9: Eigenvalues and Diagonalisation

• Eigenvalue and eigenvector: a vector whose direction a matrix preserves, with the scaling factor as eigenvalue.
• Diagonalisation: writing a matrix as a diagonal one in a basis of eigenvectors, which makes matrix powers and linear difference and differential systems tractable.

Chapters 10–13: Orthogonality and Complex Spaces

• Gram–Schmidt process: turning a basis into an orthonormal one.
• Quadratic form: an expression in the variables whose structure is revealed by orthogonal diagonalisation.
• Orthogonal projection and least squares: the closest vector in a subspace, used to fit functions to data.
• Spectral decomposition: the expression of a normal matrix through its eigenvalues and eigenvectors.

Critical Analysis

Strengths

• A stated priority on understanding. The preface says the aim is to help students not only to do linear algebra but to understand it, so that they know why the methods work. The chapter structure carries this out: reading prompts, learning outcomes and a review of each chapter's main points. The zbMATH reviewer, Peter P. Zabreĭko, calls the text clear, accessible to beginners and well balanced.
• Built-in self-study support. Comments on the activities follow each chapter, and the exercises have full solutions, not just answers, at the end of the book (the back section of about eighty pages). The preface advises attempting an exercise seriously before reading its solution. The publisher describes the book as suited to both course use and self-study.
• Broad enough for a second course. Besides the standard first-year topics, the book includes quadratic forms, least-squares fitting, unitary diagonalisation and spectral decomposition. The preface says some material is more advanced than a typical first-year course.
• Written for non-specialists. The authors teach at the London School of Economics and wrote parts of the book from self-study materials for distance-learning programmes in economics, management, finance and the social sciences.

Limitations

• Matrices before abstraction. Abstract vector spaces appear only in Chapter 5, after about 150 pages of matrix and Rⁿ material. Readers who want the subject organised around vector spaces and linear maps from the start will find another book a better fit.
• A concepts-and-methods balance, not a proof-first treatment. The preface describes the style as open and sometimes conversational, with a "dash of abstraction". Students heading to a proof-based algebra course may want a more abstract second pass.
• Topics outside the contents. The table of contents lists no singular value decomposition, Jordan form or numerical methods, so the book is not a route to those topics.
• Unsolved problems. The end-of-chapter problems have no solutions in the book; the preface says teachers can obtain them from the book's webpage.
• Scope of applications. The applications named in the contents are Leontief input–output analysis, difference and differential equations, quadratic forms and least squares, rather than a wide range of fields.

Where It Fits in the Course

This is a complete first course with a stronger emphasis on explanation than on speed. Gilbert Strang's Introduction to Linear Algebra builds the subject around matrices and the four fundamental subspaces, with applications to data and computation, and ends with the singular value decomposition that Anthony and Harvey omit. Jim Hefferon's Linear Algebra is a free text for a first course that proves its results while building mathematical maturity gently, with fully worked answers. Howard Anton's Elementary Linear Algebra is another undergraduate first-course text with applications. Of these, Anthony and Harvey are closest to Hefferon in audience and closest to Strang in treating matrices before abstract spaces.

Sheldon Axler's Linear Algebra Done Right is a proof-based treatment that starts from vector spaces and linear maps and postpones determinants; it works best as a second pass once the material here is familiar. Horn and Johnson's Matrix Analysis is a graduate reference on matrix theory and a long step beyond this book, which can serve as preparation for it.

How to Use This Book

The book suits first-year students of mathematics and of subjects that use it (economics, management, finance, the social sciences and sciences) and anyone studying alone. It assumes school algebra and trigonometry; the preliminary chapter covers what else is needed.

For a one-semester first course, read Chapters 1 to 8 in order, then Chapter 10, taking Section 3.5 (Leontief analysis) and Chapter 9 as optional applications. A second term can cover Chapter 9 and Chapters 11 to 13. Chapter 12's least-squares section pairs naturally with Chapter 10, and Chapter 13 is best left until the real theory is secure.

Work with pen and paper: do each activity when it appears, check it against the comments at the end of the chapter, then use the learning outcomes as a checklist. Attempt the exercises before reading the solutions at the back, and treat the problems as assessment practice. If the abstract material in Chapters 5 to 7 feels sudden, return to the matrix examples in Chapters 1 to 4, and consult a proof-oriented text such as Axler or Hefferon alongside it.