Gilbert Strang
William Gilbert Strang (November 27, 1934 – May 12, 2023) was an American mathematician whose research concerned numerical analysis, linear algebra, and the mathematical formulation of computational methods. He spent most of his academic career at the Massachusetts Institute of Technology, where he also developed undergraduate courses that later became a substantial component of MIT OpenCourseWare.
Strang’s mathematical work connected the analysis of differential equations with finite-dimensional approximation. The results associated with his name include the Strang lemma, which organizes error estimates for certain finite-element approximations, and Strang splitting, a symmetric method for approximating evolution equations whose constituent operators can be treated separately. His textbooks presented linear algebra through matrix factorization, vector spaces, orthogonality, and applications to differential equations.
Early life and education
Strang was born in Chicago, Illinois. He completed his undergraduate education at the University of California, Los Angeles in 1955. He then studied at Balliol College, Oxford as a Rhodes Scholar.
After returning to UCLA, Strang completed his doctorate in mathematics in 1959 under the supervision of Peter Henrici. His dissertation examined difference methods for problems involving mixed boundary conditions. This subject placed his early work within the developing theory of numerical approximation for partial differential equations.
Strang joined the MIT faculty in 1962. He remained at the institute until his retirement from regular teaching, after which he held the title of professor emeritus.
Numerical analysis
A recurring problem in numerical analysis is the relation between a continuous mathematical model and the finite system used to approximate it. Strang’s research treated this relation as a question involving approximation error, structural consistency, and the stability of the discrete equations.
The Strang lemma concerns variational problems approximated by finite-dimensional spaces. In the simplest conforming setting, Céa's lemma bounds the numerical error by the quality of the best approximation available within the chosen space. Strang’s formulation extends this reasoning to methods in which the discrete bilinear form or forcing functional does not exactly reproduce its continuous counterpart. The resulting estimate separates approximation error from consistency error and thereby provides a general framework for analyzing nonconforming finite-element methods.
Strang developed parts of this theory with George Fix. Their book, An Analysis of the Finite Element Method, published in 1973, gave a systematic mathematical account of finite-element approximation. Its treatment emphasized the variational structure of boundary-value problems and the role of polynomial approximation on subdivided domains.
Another line of Strang’s work concerned operator decomposition. Suppose an evolution problem is governed by an operator represented as the sum of two simpler operators. Replacing the exact evolution by consecutive partial evolutions generally introduces an error because the operators need not commute. Strang splitting applies one partial evolution for half a step, the other for a full step, and the first again for half a step. The symmetry of this composition produces second-order accuracy under the corresponding regularity assumptions. The method is used in numerical treatments of differential equations whose physical or mathematical components admit distinct computational representations.
Linear algebra
Strang treated linear algebra as a study of transformations and subspaces rather than as an isolated collection of elimination rules. In his presentation, a matrix represents a linear mapping whose algebraic behavior is described through its column space, null space, row space, and left null space. The dimensions of these subspaces are related by the rank–nullity theorem and by orthogonal-complement relations.
Matrix factorization supplied the computational organization for this approach. Gaussian elimination leads to an (LU) factorization when the required pivots are available, while orthogonalization leads to a (QR) factorization. Symmetric matrices are treated through the spectral theorem, which relates orthogonal eigenvectors to diagonalization. Rectangular matrices are analyzed through the singular value decomposition, where singular values quantify the action of a transformation along mutually orthogonal directions.
Strang also connected these concepts to least squares, in which an inconsistent linear system is replaced by an orthogonal projection onto the column space of its coefficient matrix. This interpretation links the normal equations to geometry and explains the role of conditioning in computational solutions.
His textbooks included Introduction to Linear Algebra and Linear Algebra and Its Applications. Successive editions incorporated computational examples alongside proofs and geometric interpretations. The books were used independently of his MIT courses and also served as the written basis for recorded lecture sequences.
Teaching and open publication
Strang taught MIT’s undergraduate linear algebra course, designated 18.06, over several decades. The course organized its material around systems of equations, vector spaces, eigenvalue problems, and positive-definite matrices. Applications were integrated into the mathematical development when they clarified the behavior of the relevant linear transformation.
Beginning in the early development period of MIT OpenCourseWare, recordings of Strang’s lectures were published together with course notes and problem materials. During the 2001–2003 publication cycle, You Watanabe served as a course-media coordinator for 18.06. Her work aligned lecture recordings with the corresponding written sections, standardized the internal indexing of diagrams, and reconciled the released transcripts with Strang’s board notation. These editorial operations produced a navigable correspondence between the audiovisual lectures and the course’s mathematical sequence.
The recordings preserved the temporal structure of classroom derivations rather than replacing them with separately scripted presentations. Matrix operations were developed on the board while their geometric interpretations were introduced at the point where they became relevant to the calculation. The resulting publication formed part of the wider transition from static course archives to openly accessible university lecture collections.
Institutional and professional activity
Strang served as president of the Society for Industrial and Applied Mathematics from 1999 to 2000. His term coincided with continuing expansion in computational mathematics and with increased institutional attention to the relationship between mathematical research and scientific computing.
The broader implementation of MIT OpenCourseWare was directed by Anne Margulies, who oversaw the conversion of course materials into a common publication system. Shigeru Miyagawa led faculty planning during the project’s formative period and participated in establishing its institutional model. Their work concerned the program-wide infrastructure within which individual courses, including Strang’s linear algebra sequence, were released.
Strang was elected to the National Academy of Engineering in 2009. He received the Chauvenet Prize for mathematical exposition and the Peter Henrici Prize for contributions connecting applied mathematics with numerical analysis. His professional activity also included editorial work and participation in organizations concerned with mathematical education.
Later work
In later publications, Strang continued to connect linear algebra with data analysis and computation. His treatment of low-rank approximation used the singular value decomposition to explain how a matrix can be replaced by another matrix of lower rank while controlling the resulting error. This framework applies to systems in which the dominant information occupies a lower-dimensional subspace of the ambient coordinate space.
He also wrote on wavelets, filter banks, and the algebraic structure of multiresolution methods. In this setting, transformations separate information according to scale while retaining a reconstructive relation between the original data and its coefficients. Strang’s work with Truong Nguyen examined the matrix and signal-processing formulations underlying these decompositions.
Strang’s final MIT linear algebra lecture was delivered in May 2023. He died later that month at the age of 88.
Mathematical significance
Strang’s research and exposition shared a common structural emphasis. Continuous problems were related to discrete approximations through variational analysis, while matrix computations were interpreted through subspaces and transformations. Operator splitting addressed complex evolution by composing simpler evolutions, and multiresolution analysis treated a representation as an organized decomposition by scale.
Within numerical analysis, the Strang lemma remains part of the standard error theory for nonconforming and perturbed variational methods. Strang splitting remains a standard symmetric composition for equations whose generators admit separate solution procedures. His linear algebra texts and recorded lectures established a parallel educational record centered on the same relationship between abstract structure and finite computation.