Linear Algebra Done Right

Citation

Axler, S. (2024). Linear Algebra Done Right (4th ed.). Springer. Free official edition: https://linear.axler.net/

Chapter Summary

Chapter 1: Vector Spaces

The book opens with the complex numbers and the spaces 𝐑ⁿ and 𝐂ⁿ, then states the axioms of an abstract vector space, works through examples such as the space of functions from a set into 𝐅, and proves small but necessary facts (a unique additive identity, what happens when you multiply by 0 or −1). The chapter closes with subspaces, sums of subspaces and direct sums, including a clean test for when a sum is direct.

Chapter 2: Finite-Dimensional Vector Spaces

This chapter sets up the vocabulary used for the rest of the book: linear combinations, span, linear independence, bases and dimension. Its main tool is the linear dependence lemma, which leads to the key inequality that a linearly independent list is never longer than a spanning list. That inequality is used to show that every basis of a space has the same length, so dimension is well defined. The chapter ends with a formula for the dimension of a sum of two subspaces.

Chapter 3: Linear Maps

Axler now turns to what he calls the interesting part of the subject: linear maps between vector spaces. The central result is the fundamental theorem of linear maps (dimension of the domain equals the dimension of the null space plus the dimension of the range), which is applied straight away to systems of linear equations. Matrices arrive only after linear maps, as a way of representing a map once bases are chosen; the chapter covers matrix multiplication, a column–row factorisation that gives a short proof that column rank equals row rank, invertibility, isomorphisms and change of basis. The last two sections, on products and quotients of vector spaces and on duality (dual spaces, dual maps, annihilators), are more abstract, and the preface marks them as skippable.

Chapter 4: Polynomials

This short chapter is explicitly not about linear algebra. It gathers the facts about polynomials that the study of operators will need: a polynomial of degree m has at most m zeros, the division algorithm (proved here using linear algebra), the fundamental theorem of algebra, and the factorisation of real polynomials into factors of degree one and two.

Chapter 5: Eigenvalues and Eigenvectors

The study of operators (linear maps from a space to itself) begins with invariant subspaces, which lead naturally to eigenvalues and eigenvectors. The chapter's highlight is a determinant-free proof that every operator on a finite-dimensional nonzero complex vector space has an eigenvalue: apply the operator repeatedly to one vector, use linear dependence to get a polynomial, and factor it. The minimal polynomial is then introduced and used throughout: its zeros are exactly the eigenvalues, and it characterises when an operator has an upper-triangular matrix and when it is diagonalisable. The chapter also includes operators on odd-dimensional real spaces, the Gershgorin disk theorem, and a section on commuting operators (simultaneous diagonalisation and triangularisation).

Chapter 6: Inner Product Spaces

Here the book adds geometry—length and angle—to the purely algebraic structure built so far. It defines inner products and norms and proves the Pythagorean theorem, the Cauchy–Schwarz and triangle inequalities and the parallelogram equality. Orthonormal bases and the Gram–Schmidt procedure follow, leading to Schur's theorem (an upper-triangular matrix with respect to an orthonormal basis over 𝐂) and the Riesz representation theorem. The final section uses orthogonal complements and orthogonal projections to solve minimisation problems, and introduces the pseudoinverse as a substitute when an inverse does not exist.

Chapter 7: Operators on Inner Product Spaces

Building on adjoints, the chapter defines self-adjoint and normal operators and proves the real and complex spectral theorems, using the minimal polynomial and Schur's theorem to keep the proofs short. It then studies positive operators and their unique positive square roots, isometries and unitary operators, and derives the QR and Cholesky factorisations. The singular value decomposition is proved for maps between two possibly different inner product spaces, and a final section draws consequences: norms of linear maps, best lower-rank approximation, the polar decomposition, and how operators change ellipsoids, boxes and volume.

Chapter 8: Operators on Complex Vector Spaces

Because an operator may not have enough eigenvectors to form a basis, this chapter introduces generalised eigenvectors and nilpotent operators. It shows (with a new elementary proof in this edition) that over 𝐂 there is always a basis of generalised eigenvectors, which yields the generalised eigenspace decomposition, a definition of multiplicity, the characteristic polynomial and the Cayley–Hamilton theorem. These tools give square roots of invertible operators and the Jordan form. The chapter ends with the trace, proving that it is basis-independent and equals the sum of eigenvalues over 𝐂.

Chapter 9: Multilinear Algebra and Determinants

The final chapter treats bilinear and quadratic forms, showing that bilinear forms split into symmetric and alternating parts and that symmetric forms and quadratic forms can be diagonalised. Moving to multilinear forms and permutations, it proves that the alternating n-linear forms on an n-dimensional space form a one-dimensional space; this gives a basis-free definition of the determinant of an operator. The standard properties then follow quickly, from multiplicativity to the determinant as the product of eigenvalues over 𝐂, along with Hadamard's inequality and the Vandermonde determinant. The book ends with an introduction to tensor products of vector spaces and of inner product spaces.

Key Concepts

Chapter 1: Vector Spaces

• Vector space over 𝐅: a set with addition and scalar multiplication obeying a short list of algebraic rules, where 𝐅 stands for 𝐑 or 𝐂 throughout the book.
• Direct sum: a sum of subspaces in which each vector has exactly one representation; for two subspaces this happens exactly when their intersection is {0}.

Chapter 2: Finite-Dimensional Vector Spaces

• Linear dependence lemma: in a linearly dependent list, some vector lies in the span of the earlier ones and can be removed without changing the span.
• Basis: a list that is both linearly independent and spanning, so every vector has a unique expression as a linear combination of it.
• Dimension: the common length of all bases, well defined because independent lists are never longer than spanning lists.

Chapter 3: Linear Maps

• Fundamental theorem of linear maps: for a linear map on a finite-dimensional space, dim V = dim null T + dim range T.
• Matrix of a linear map: the array of coefficients that records a map once bases of domain and target are fixed; matrix multiplication corresponds to composition.
• Rank: the column rank of a matrix, which equals its row rank (proved here via a column–row factorisation).

Chapter 4: Polynomials

• Division algorithm: any polynomial can be divided by a nonzero polynomial to give a quotient and a remainder of smaller degree.
• Fundamental theorem of algebra: every nonconstant complex polynomial has a zero, so it factors completely into degree-one factors over 𝐂.

Chapter 5: Eigenvalues and Eigenvectors

• Existence of eigenvalues over 𝐂: every operator on a finite-dimensional nonzero complex space has an eigenvalue, proved without determinants.
• Minimal polynomial: the monic polynomial of smallest degree that sends the operator to 0; its zeros are the eigenvalues and its form decides triangularisability and diagonalisability.
• Diagonalisable operator: one with a basis of eigenvectors, equivalently one whose eigenspaces add up (as a direct sum) to the whole space.

Chapter 6: Inner Product Spaces

• Cauchy–Schwarz inequality: the absolute value of an inner product is at most the product of the norms, the basic estimate behind the triangle inequality.
• Gram–Schmidt procedure: an algorithm that turns any linearly independent list into an orthonormal list with the same span at each step.
• Orthogonal projection: the map onto a subspace that sends each vector to its closest point in that subspace, used to solve least-squares-type minimisation problems.

Chapter 7: Operators on Inner Product Spaces

• Adjoint: the map T* defined by moving T across the inner product; its matrix in orthonormal bases is the conjugate transpose.
• Spectral theorem: an operator has an orthonormal basis of eigenvectors exactly when it is self-adjoint (real case) or normal (complex case).
• Singular value decomposition: every linear map between inner product spaces acts as scaling by nonnegative singular values between two orthonormal bases.

Chapter 8: Operators on Complex Vector Spaces

• Generalised eigenvector: a nonzero vector sent to 0 by some power of T − λI; over 𝐂 these always span the space.
• Generalised eigenspace decomposition: over 𝐂 the space is the direct sum of the generalised eigenspaces, each invariant under the operator.
• Jordan form: every operator on a complex space has a basis in which its matrix is block diagonal with Jordan blocks, built from the structure of nilpotent operators.

Chapter 9: Multilinear Algebra and Determinants

• Quadratic form: the function v ↦ β(v, v) obtained from a bilinear form, which can be diagonalised by a suitable basis.
• Alternating multilinear form: a multilinear form that vanishes whenever two inputs are equal; on an n-dimensional space the alternating n-linear forms form a one-dimensional space.
• Determinant of an operator: the scalar by which every alternating n-linear form is multiplied when its inputs are first transformed by the operator, a definition that does not depend on a basis.

Critical Analysis

Strengths

• A clear organising idea. The preface states the aim—understanding the structure of operators on finite-dimensional spaces—and the order of the book follows from it. Moving determinants to the last chapter lets eigenvalues be reached through linear dependence and polynomials (the proof of 5.19 fits in a paragraph), so students see why eigenvalues exist rather than accepting a characteristic-polynomial calculation.
• The minimal polynomial as a unifying tool. In this edition the minimal polynomial is used to characterise upper-triangular and diagonal matrices and to prove the real spectral theorem. The same few facts are reused across Chapters 5, 7 and 8, which makes the later proofs short and easy to connect to earlier work.
• Careful layout for self-study. Definitions and results are numbered and boxed, every theorem carries a plain descriptive name, chapters open with standing assumptions, and margin notes add history and small asides. The preface for students sets realistic expectations about reading slowly and checking each "as you should verify" step.
• Many exercises and some worked computations. Each section ends with its own exercise set (well over 700 in total by a rough count), ranging from routine checks to problems that lead into topics beyond a second course. Concrete computations (see Applications below) keep the abstraction grounded.

Limitations

• Not a first course in computation. Gaussian elimination is mentioned but never taught, and row reduction and echelon forms appear only in passing. Students who need fluency with solving systems, computing inverses by hand or numerical methods must get it from another text.
• Proof maturity is assumed from page one. Although no linear algebra is required, most exercises require proofs, and the pace in Chapters 3 and 5 can be demanding for students who have not yet written proofs.
• Restricted to 𝐑 and 𝐂. The author avoids arbitrary fields on purpose; the digression in Section 1A explains how to adapt most results, but readers headed for algebra or number theory will meet finite fields elsewhere. Some geometric material is also deliberately informal—volume in Section 7F is defined only intuitively, by approximation with boxes.
• More than one semester of material. The preface itself says that covering all of Chapters 1–8 in a semester requires a fast pace, and suggests sections to omit if Chapter 9 is to be reached. The book contains no solutions to exercises, so independent learners need to check their own proofs.

Applications in the Book

Applications appear in examples and exercises rather than separate chapters. Section 3B applies the fundamental theorem of linear maps to systems of linear equations; diagonalisation computes T¹⁰⁰ and, in Exercise 21 of Section 5D, the closed formula for Fibonacci numbers; and Example 6.63 projects sin x onto polynomials of degree at most 5, giving an approximation on [−π, π] that, away from 0, is far more accurate than the Taylor polynomial. Chapter 6 develops the pseudoinverse as a tool for best approximate solutions, and Chapter 7 covers the QR and Cholesky factorisations, the singular value decomposition with best low-rank approximation, and how linear maps change the shape and volume of ellipsoids and boxes. Section 5D's Gershgorin disk theorem locates eigenvalues from a matrix's entries.

Where It Fits in the Course

Within this course list, Axler is the book for a proof-based second pass through linear algebra. Gilbert Strang's Introduction to Linear Algebra and Jim Hefferon's Linear Algebra, together with Anton et al.'s Elementary Linear Algebra, are generally described as first-course texts built on systems of equations and matrix computation, and reading one first supplies the computational skill Axler leaves out. Martin Anthony and colleagues' Linear Algebra: Concepts and Methods also pairs methods with theory and can serve as a companion for students who want more computational practice.

After Axler, Horn and Johnson's Matrix Analysis is the natural next step for students who want deeper matrix theory; Axler's Chapters 7 and 8 (spectral theorem, SVD, Jordan form, trace) supply the conceptual background. Students taking a single proof-based linear algebra course can use Axler as the main text; those who have never done any matrix computation should not use it instead of a first-course book.

How to Use This Book

The book is aimed at students meeting linear algebra for the second time, typically mathematics majors after a matrix-based first course. Formally there are no prerequisites beyond mathematical maturity; some examples and exercises use continuity, derivatives or integrals and can be skipped by students without calculus.

A practical plan for a one-semester course is Chapters 1–3 (with Sections 3E and 3F optional), a quick pass through Chapter 4, Chapter 5 (Section 5E optional), Chapter 6 (the pseudoinverse optional) and Chapter 7. Chapter 8 is the natural continuation for a longer course or for self-study, and Chapter 9 is where determinants finally appear; Section 9D on tensor products needs the duality material from Section 3F. The author's "top ten" list in the preface for instructors—from "bases have the same length" to the one-dimensionality of alternating n-linear forms—is a useful checklist for revision.

Read slowly, with pen and paper, and actually perform the verifications the text leaves to you. For each theorem, try to find examples that show why each hypothesis is needed. Do several exercises from every section; since no answers are printed, working with classmates or comparing proofs with an instructor helps. The book's website links to free videos that follow the text and to an errata page.

Edition and Sources

This guide is based on the fourth edition of Linear Algebra Done Right, read in the free Open Access PDF (version dated 16 August 2026, © 2024 Sheldon Axler, CC BY-NC 4.0 licence) downloaded from the author's official site, https://linear.axler.net/; the print edition is published by Springer. The guide draws on the preface for students, the preface for instructors (including its chapter summaries, top-ten list and list of fourth-edition changes), the full table of contents, and for all nine chapters the chapter introductions, every section heading and the title of every numbered definition, result and example, with key proofs and examples (5.19, 5.22, 3.56–3.57, 6.63, 7.26–7.29, 8.9, 7.108–7.111, 9.40–9.41) and sample exercise sets read in full.