Introduction to Linear Algebra

Citation

Strang, G. (2023). Introduction to Linear Algebra (6th ed.). Wellesley-Cambridge Press.

Chapter Summary

The sixth edition (copyright 2022, published 2023) has ten chapters, ten short appendices and indexes of equations and notation. The preface describes it as the textbook for MIT's course 18.06, whose video lectures on MIT OpenCourseWare accompany the book, and explains the main change from earlier editions: the course now starts with the columns of a matrix, linear independence and the column space, introducing the factorization A = CR in Chapter 1 instead of beginning with systems of equations. Two organising ideas run through the book, the four fundamental subspaces and five factorizations of a matrix (A = CR, LU, QR, S = QΛQᵀ and the singular value decomposition A = UΣVᵀ). The preface says the first seven chapters "more than fill up most linear algebra courses", followed by optional chapters leading to optimization and deep learning. The sample sections on the book's website open with the section's key ideas in a box and end with a problem set; some also have a "Review of the Key Ideas", worked examples and challenge problems.

Chapter 1: Vectors and Matrices

Vectors and linear combinations, lengths and angles from dot products, matrices and their column spaces, and matrix multiplication AB and CR. The preface's opening example builds up a 3 by 4 matrix one column at a time to show independent and dependent columns and how C and R record them.

Chapter 2: Solving Linear Equations Ax = b

Elimination and back substitution, elimination matrices and inverse matrices, matrix computations and A = LU, permutations and transposes, and a closing section on derivatives and finite difference matrices that links matrices to calculus.

Chapter 3: The Four Fundamental Subspaces

Vector spaces and subspaces, computing the nullspace by elimination with A = CR, the complete solution to Ax = b, independence, basis and dimension, and the dimensions of the four subspaces. The sample Section 3.5 proves that row space and column space both have dimension r, the rank, and illustrates the left nullspace with the incidence matrix of a graph and Kirchhoff's Current Law.

Chapter 4: Orthogonality

Orthogonal vectors and subspaces, projections onto lines and subspaces, least squares approximations, orthonormal bases and Gram–Schmidt (A = QR), and the pseudoinverse of a matrix.

Chapter 5: Determinants

A short chapter: 3 by 3 determinants and cofactors, computing and using determinants, and areas and volumes by determinants.

Chapter 6: Eigenvalues and Eigenvectors

An introduction to Ax = λx, diagonalizing a matrix, symmetric positive definite matrices, complex numbers, vectors and matrices, and solving linear differential equations.

Chapter 7: The Singular Value Decomposition (SVD)

Singular values and singular vectors, image processing by linear algebra, and principal component analysis by the SVD. The preface urges readers who learn eigenvalues in Chapter 6 to continue to Section 7.1, and notes that the SVD completes the Fundamental Theorem of Linear Algebra by giving special bases for all four subspaces.

Chapter 8: Linear Transformations

The idea of a linear transformation, the matrix of a linear transformation, and the search for a good basis.

Chapter 9: Linear Algebra in Optimization

New to the edition: minimizing a multivariable function, backpropagation and stochastic gradient descent, constraints, Lagrange multipliers and minimum norms, and linear programming, game theory and duality. The preface presents this as "where linear algebra meets calculus".

Chapter 10: Learning from Data

Also new: piecewise linear learning functions, creating and experimenting, and mean, variance and covariance.

Appendices

Ten short appendices: the ranks of AB and A + B, a summary of matrix factorizations, counting parameters in the basic factorizations, codes and algorithms for numerical linear algebra, the Jordan form, tensors, the condition number, Markov matrices and Perron–Frobenius, elimination and factorization, and computer graphics.

Key Concepts

Chapters 1–3: Columns, elimination and subspaces

• Column space: all linear combinations of the columns of A.
• A = CR: a matrix as its independent columns C times the matrix R that combines them to produce every column.
• Elimination and A = LU: solving Ax = b by row operations, recorded as lower triangular times upper triangular.
• Four fundamental subspaces: column space, row space, nullspace and left nullspace, with dimensions r, r, n − r and m − r.
• Rank: the number of independent columns, equal to the number of independent rows.

Chapters 4–6: Orthogonality, determinants and eigenvalues

• Projection and least squares: the closest solution when Ax = b has none, found by projecting b onto the column space.
• Gram–Schmidt and A = QR: producing orthonormal columns.
• Eigenvalues and diagonalization: vectors that A only stretches, and A = XΛX⁻¹ when there are enough of them.
• Symmetric matrices and S = QΛQᵀ: real eigenvalues and orthonormal eigenvectors; positive definite matrices.

Chapters 7–10: SVD, transformations and data

• Singular value decomposition: A = UΣVᵀ for every matrix, with orthonormal singular vectors and positive singular values.
• Principal component analysis: finding the main directions of variation in data from the SVD.
• Linear transformation and change of basis: the matrix of a transformation depends on the basis; a good basis simplifies it.
• Gradient descent and backpropagation: the optimization methods behind training learning functions.

Critical Analysis

Strengths

• A clear organising picture. The four subspaces and the five factorizations recur chapter after chapter, so students always know where a new idea fits; the factorizations are printed on the cover.
• Columns and independence from the start. Introducing column spaces and A = CR in Chapter 1 gives students the central ideas of rank and independence before the mechanics of elimination, the change the preface highlights.
• Applications built in. Sections on finite differences, graphs and Kirchhoff's law, least squares, image processing, PCA, differential equations, optimization and learning from data connect the theory to engineering and data science.
• Exceptional free support. The book is coordinated with the MIT 18.06 video lectures on OpenCourseWare; the official website offers sample sections, selected solutions to the problem sets for Chapters 1–8, practice exams from MIT, errata and sections from the fifth edition that moved online.

Limitations

• Not proof-centred. The style is explanatory and computational; formal proofs and abstract vector spaces play a smaller role than in a theory-first course.
• Topics moved to the web. The preface notes that Fourier series, norms and condition numbers, iterative methods and cryptography left the printed book for the website; Allen Stenger's MAA Reviews notice (April 2023) also reports that many worked examples and challenge problems from the fifth edition were dropped.
• Determinants compressed. Chapter 5 has only three sections, so courses that emphasise determinant theory need supplementary material.
• Informal exposition. Strang writes conversationally, in the first person ("Here is the key!"); key results are boxed, but students revising for exams may want to compile their own list of definitions and theorems.

Where It Fits in the Course

In this course Strang is the applied, matrix-centred first text. Jim Hefferon's free Linear Algebra covers a first course with more emphasis on proof and fully worked answers to its exercises, and suits students who want to build mathematical maturity; Howard Anton's Elementary Linear Algebra is another standard first-course textbook. Students can use either alongside Strang for a different explanation of the same topics.

After Strang, Sheldon Axler's Linear Algebra Done Right develops the subject through vector spaces and linear maps with proofs, postponing determinants, the natural second course for mathematics students. Martin Anthony and M. Harvey's Linear Algebra: Concepts and Methods offers a rigorous treatment with applications, and Horn and Johnson's Matrix Analysis is the graduate reference on matrix theory for those continuing from Strang's Chapters 6–7.

How to Use This Book

Secondary-school algebra is the main prerequisite; some calculus helps for Sections 2.5 and 6.5 and Chapter 9.

• A one-semester course: Chapters 1–7 in order. Chapters 1 and 3 carry the central ideas of column space and the four subspaces; Section 3.5 and Chapter 4 on projections and least squares are essential; Chapter 5 can be covered quickly.
• Do not stop at Chapter 6: the preface asks readers to continue to Section 7.1 on the SVD, which ties the course together.
• Optional chapters: Chapter 8 for a more abstract view, Chapters 9–10 for students interested in optimization, machine learning and statistics.
• Use the lectures: watch the corresponding 18.06 lecture before or after each section; the website's selected solutions and old MIT exams provide practice with feedback.
• Work the problem sets, starting with the worked examples and moving to the challenge problems once the basic problems are easy.

Students with the fifth edition should note that sections were reorganised and some topics moved online.