Strang, Gilbert

professor of mathematics at MIT; author of "Introduction to Linear Algebra"; creator of MIT's widely-viewed OpenCourseWare linear algebra lectures

Gilbert Strang: Linear Algebra, Teaching, and MIT OpenCourseWare

Lex Fridman Podcast · Nov 25, 2019

• Gilbert Strang
• linear algebra
• MIT OpenCourseWare
• singular value decomposition
• four fundamental subspaces
• deep learning and matrices
• piecewise-linear functions
• finite element method
• math education
• calculus vs. linear algebra

Core ideas

Strang recounts how MIT's OpenCourseWare project began: a committee tasked by then-president Charles Vest with figuring out how MIT could commercially package its courses instead concluded, after a weekend of reflection, that MIT should simply give the material away for free — a decision Strang frames as capturing something essential about MIT's institutional identity, since the courses (his own 18.06 linear algebra lectures among them) demonstrate the actual texture of MIT teaching rather than an abstracted brand. He credits linear algebra's own rising importance, more than any special quality of his teaching, for the videos' massive viewership, tracing this rise directly to the explosion of data science: because real-world data naturally arrives in matrix form (e.g., a matrix where each column is a drug and each row a patient), linear algebra — not calculus or differential equations — has become the mathematical toolkit best suited to describing and manipulating it.

He walks through the four fundamental subspaces of a matrix (column space, row space, and their two orthogonal null spaces) as one of the ideas he's proudest of communicating clearly, and generalizes his approach to visualization: despite readily admitting he has no real mental image of a vector in ten dimensions, he insists this is not an obstacle, since the algebraic operations (addition, scalar multiplication) generalize identically from 3D to 10D or higher without requiring visualization — describing his own habit of picturing a flat plane in high-dimensional space by analogy to 3D geometry, in the same spirit as viewers who can't visualize a vector but can still compute with one. He identifies the Singular Value Decomposition — the theorem that any matrix, however irregular, can be factored into rotation × stretch × rotation — as an especially beautiful result precisely because it reduces an opaque grid of numbers into three components (two rotations and a diagonal "stretch" matrix) that are each intuitively visualizable, even in high dimensions, unlike eigenvalues, which only apply to square matrices and can't handle the rectangular matrices that most real data comes in.

On deep learning, Strang describes it as fundamentally a data-driven search for a rule mapping known inputs to known outputs, so that the rule can then be applied to new, unseen inputs — a different modern approach to problems traditionally solved by finding governing physical laws (Newton, Einstein) directly. He explains the mathematical core of a neural network as combining flat, linear pieces (matrices) with a nonlinear "fold" — the simplest possible nonlinearity being a piecewise-linear function with one slope on one side and a different slope on the other — arguing that a huge number of these folds, stacked and combined, can approximate remarkably complex, near-curved surfaces despite every individual piece remaining flat; he compares this directly to the finite element method long used by engineers to model bridges, buildings, and airplanes via piecewise-flat approximations of curved structures. He stresses that this kind of learning only works if there is some real underlying signal or pattern in the data to find — purely random data offers nothing for a network to discover — framing "learning the rules" via data as a genuinely new complement to the older scientific tradition of deriving rules from first principles.

Strang stakes out an explicit, self-described "biased" curricular opinion: that university mathematics education overemphasizes calculus at the expense of linear algebra, arguing linear algebra is inherently more tractable for students (since nothing "bends" — every relevant surface is flat, so ideas that work in 3D generalize cleanly to 10D or 100D) and has become dramatically more practically important than calculus in the current data-driven era, alongside probability and statistics as the third major branch he believes deserves comparably greater curricular emphasis. Asked about his personal favorite matrix, he names the tridiagonal matrix with 2s on the main diagonal and −1s on the diagonals immediately above and below it, which he connects to the linear-algebra analogue of the second derivative — a discrete approximation of curvature that recurs constantly across engineering applications.

On teaching itself, Strang candidly admits he dislikes the assessment side of the job (exams and grading) and identifies primarily with the moment of first excitement when a student encounters a new idea — describing his goal as helping students feel that a subject "has some beauty" rather than optimizing for evaluation, and locating himself on the spectrum of mathematical work between pure theorists and applied scientists as someone drawn to structure itself (the relationships between a matrix's rows and columns) rather than to physics, chemistry, or neuroscience applications directly. Asked about STEM representation in politics, he laments the near-total absence of engineers or mathematicians among elected officials, briefly recalling testifying before a Congressional committee around 2000 during his tenure as president of the Society for Industrial and Applied Mathematics, when math briefly received unusual attention in Washington.

Excerpts