Linear Algebra

Citation

Hefferon, J. (2020). Linear Algebra (4th ed.). Orthogonal Publishing L3C. Free official edition: https://hefferon.net/linearalgebra/

Chapter Summary

Chapter One: Linear Systems

The book opens with two small applied systems (balancing masses and a chemical reaction) and then develops Gauss's Method, proving that the row operations it uses do not change the solution set. It describes solution sets in vector form and proves that every solution set is a particular solution plus the solutions of the associated homogeneous system, so it is empty, a single point or infinite. A section on linear geometry in ℝⁿ (lines, planes, length, angle, the Cauchy–Schwarz and triangle inequalities) is starred as optional. The chapter ends with Gauss–Jordan reduction, the Linear Combination Lemma, and the proof that every matrix reduces to a unique reduced echelon form. Topics cover computer algebra systems, input–output analysis, accuracy of computations and electrical networks.

Chapter Two: Vector Spaces

Motivated by the observation that solution sets of homogeneous systems are closed under linear combinations, this chapter defines a vector space over the reals and works through many examples (including spaces of polynomials, matrices and functions) and subspaces and spans. It then studies linear independence, bases and dimension, using the Exchange Lemma to prove that all bases of a finite-dimensional space have the same size. Section III.3 connects these ideas back to linear systems through row space, column space and the theorem that row rank equals column rank; an optional section treats sums and direct sums of subspaces. Topics: fields, crystals, voting paradoxes and dimensional analysis.

Chapter Three: Maps Between Spaces

The longest chapter makes precise the idea that some spaces are "the same": isomorphisms are defined, and dimension is shown to characterise a finite-dimensional space up to isomorphism. Homomorphisms (linear maps) follow, with range space, null space and the theorem that rank plus nullity equals the dimension of the domain. Matrices are then introduced as representations of linear maps relative to bases, so that matrix multiplication is defined to represent composition; inverses, change of basis, and the result that matrices are equivalent exactly when they have the same rank complete the core. Optional sections cover orthogonal projection, Gram–Schmidt orthogonalisation and projection into a subspace. Topics: line of best fit, the geometry of linear maps, magic squares, Markov chains and orthonormal matrices.

Chapter Four: Determinants

Starting from the goal of a formula that decides whether a square matrix is nonsingular, the chapter first explores small cases, then defines the determinant through the properties it must have under row operations, proves it is unique, and constructs it with the permutation expansion (the existence proof via inversions and signum is starred). A section interprets the determinant as the factor by which a map changes the volume of boxes, and an optional section gives Laplace's expansion and the adjoint (adjugate) formula for the inverse. Topics: Cramer's rule, the speed of calculating determinants, Chiò's method, projective geometry and computer graphics.

Chapter Five: Similarity

The final chapter asks for the simplest matrix that represents a transformation of a space to itself. It switches to complex scalars after a starred review of polynomial factoring and complex numbers, then defines similarity, diagonalisability, eigenvalues, eigenvectors, the characteristic polynomial and eigenspaces, proving that eigenvectors for distinct eigenvalues are independent. Optional sections study nilpotent maps through strings of basis vectors, the minimal polynomial, the Cayley–Hamilton theorem and Jordan canonical form. Topics: the method of powers, stable populations, page ranking, linear recurrences and coupled oscillators.

Appendix

A short appendix reviews statements, quantifiers, techniques of proof (including induction) and sets, functions and relations, as background for readers new to proofs.

Key Concepts

Chapter One: Linear Systems

• Gauss's Method: using row swaps, rescalings and row combinations to bring a system to echelon form without changing its solutions.
• General = particular + homogeneous: every solution set of a linear system is one particular solution plus all solutions of the related homogeneous system.
• Reduced echelon form: the unique fully reduced matrix to which a given matrix is row equivalent, so row-equivalence classes can be identified by it.

Chapter Two: Vector Spaces

• Vector space: a set with addition and scalar multiplication satisfying a list of conditions, so that results about linear combinations apply everywhere at once.
• Linear independence: no vector in the set is a linear combination of the others.
• Basis and dimension: a linearly independent spanning set, whose size is the same for every basis of a finite-dimensional space.
• Row rank equals column rank: the dimensions of the row space and column space of any matrix agree.

Chapter Three: Maps Between Spaces

• Isomorphism: a one-to-one, onto linear map; finite-dimensional spaces are isomorphic exactly when they have the same dimension.
• Rank–nullity: for a linear map, the dimension of the range plus the dimension of the null space equals the dimension of the domain.
• Matrix representation: the matrix that records a linear map relative to chosen bases; multiplying matrices represents composing maps.
• Change of basis: converting representations with a nonsingular matrix; matrices representing the same map are matrix equivalent.

Chapter Four: Determinants

• Determinant as a function: the unique function on square matrices that behaves in a prescribed way under row operations and takes the value 1 on the identity.
• Permutation expansion: a formula writing the determinant as a signed sum over permutations, which proves existence and that a matrix and its transpose share a determinant.
• Determinant as size: the absolute value of the determinant gives the factor by which a transformation scales the volume of boxes.

Chapter Five: Similarity

• Similar matrices: matrices T and PTP⁻¹ that represent the same transformation with respect to different bases.
• Eigenvalue and eigenvector: a scalar λ and nonzero vector v with t(v) = λv, found through the roots of the characteristic polynomial.
• Jordan canonical form: a nearly diagonal representation, built from nilpotent strings, that exists for every transformation of a finite-dimensional complex space.

Critical Analysis

Strengths

• A deliberate transition to proofs. The preface describes a "developmental" approach: all results are proved, but the theory is surrounded by detailed examples. Proofs begin in Chapter One—the correctness of Gauss's Method and the uniqueness of reduced echelon form—so the move to abstract vector spaces in Chapter Two does not arrive as a sudden shift.
• Linear maps before matrices. Chapter Three defines matrices as representations of maps and derives matrix multiplication from composition, so students see why the operations are defined as they are, and change of basis follows naturally.
• Large, graded exercise sets. The preface aims at about two dozen exercises per subsection, from routine checks to involved proofs, with a recommended sample marked in the margin and a few puzzles drawn from journals and competitions flagged separately. A separate answers document is available on the author's site.
• Optional applications and computing topics. Every chapter ends with short topics—input–output analysis, voting paradoxes, Markov chains, computer graphics, page ranking and others—each with exercises and pointers to further reading, short enough for a single class or a student project.

Limitations

• Real scalars, limited inner-product theory. Vector spaces are over the reals until Chapter Five; orthogonality appears only in ℝⁿ and in optional sections, and there is no general treatment of inner product spaces, the spectral theorem or the singular value decomposition.
• Advanced material is starred and compressed. Nilpotence, the minimal polynomial and Jordan form occupy optional sections at the end; a typical one-semester schedule, like the one in the preface, reaches only Chapter Five's first part on eigenvalues.
• Little numerical linear algebra. The Accuracy of Computations and Speed of Calculating Determinants topics introduce pivoting, conditioning and operation counts, but LU factorisation and iterative methods are not developed.
• Sparse notation and long chapters. Sections are cross-referenced as "Three.IV.2", and the five chapters are long, which can make navigation and finding a specific result harder than in texts with more, shorter chapters.

Applications in the Book

Applications are confined to the Topics that close each chapter. Chapter One covers computer algebra systems (with Sage examples), Leontief input–output analysis, round-off error and pivoting, and electrical and traffic networks via Kirchhoff's laws. Chapter Two treats fields, crystal lattices, voting paradoxes (decomposing majority cycles) and dimensional analysis. Chapter Three includes least-squares lines of best fit, the geometry of linear maps, magic squares, Markov chains and orthonormal matrices. Chapter Four gives Cramer's rule, operation counts, Chiò's method, projective geometry and computer graphics, and Chapter Five gives the method of powers, stable populations, page ranking, linear recurrences (such as the Fibonacci numbers) and coupled oscillators.

Where It Fits in the Course

Among the course books, Hefferon sits beside Gilbert Strang's Introduction to Linear Algebra and Anton and colleagues' Elementary Linear Algebra, which are generally described as computation-centred first courses. Hefferon covers similar ground—systems, vector spaces, maps, determinants, eigenvalues—but it proves every result and develops matrices from linear maps, so it suits students who want a first course that also builds proof skills. Students who want more matrix computation or numerical practice can pair it with one of the other first-course texts.

Horn and Johnson's Matrix Analysis is a research-level matrix theory reference and belongs after a first course; Hefferon's Chapter Five (similarity, eigenvalues and the optional Jordan form) is the most direct preparation for it, although Hefferon's limited coverage of inner products means further study of those topics is needed first.

How to Use This Book

The book is written for a standard first undergraduate course, typically for second- or third-year students with at least one semester of calculus; no previous proof writing is assumed, and the appendix reviews logic, proof techniques and set notation. The schedule in the preface assumes readers have already met the basic vector material in Section One.II.

A one-semester course can follow that schedule: Chapter One (Sections I and III), Chapter Two, Chapter Three through change of basis (Sections I–V), Chapter Four Sections I.1–I.3 and II, and Chapter Five Section II on similarity, diagonalisability and eigenvalues. Starred subsections—Jordan form, projection, Laplace expansion—are natural extensions for a longer course or self-study. Choose one or two Topics per chapter as enrichment, ideally with software such as Sage. Above all, follow the author's advice to work many exercises, especially those marked in the margin, and to justify each answer with a computation or proof before consulting the separate answers document.

Edition and Sources

This guide is based on the fourth edition of Linear Algebra by Jim Hefferon (preface dated April 2020, PDF generated May 2020), read in the free PDF linked from the author's official pages (https://hefferon.net/linearalgebra/, file hosted on the author's University of Vermont site). It draws on the notation page, the preface (approach, exercises, applications, schedule and advice), the full table of contents, all five chapter introductions, every section and subsection heading, the statements of the numbered theorems, lemmas and corollaries throughout, the opening pages of every Topic, and the appendix outline.