Gilbert Strang

Four subspaces, and the diagram that carries them

Strang’s reorganisation of the subject is one diagram. A real $m \times n$ matrix $A$ determines four subspaces: the column space $C(A)$ and the left null space $N(A^{\mathsf{T}})$ living in $\mathbb{R}^{m}$, the row space $C(A^{\mathsf{T}})$ and the null space $N(A)$ living in $\mathbb{R}^{n}$. The rank $r$ fixes all four dimensions at once — $\dim C(A) = \dim C(A^{\mathsf{T}}) = r$ — and each pair is orthogonal in its own space. Every linear map is that picture; solving, projecting, least squares and the pseudoinverse are all readings of it.

The pedagogical move is to teach a matrix by its factorisations rather than its entries. $A = LU$ is elimination. $A = QR$ is orthogonalisation. $A = Q\Lambda Q^{\mathsf{T}}$ is the spectral theorem for symmetric matrices. $A = U\Sigma V^{\mathsf{T}}$ is the singular value decomposition, which exists for every matrix with no hypotheses at all — which is exactly why it is the right instrument when you do not yet know whether your operator is well behaved. Late in his career he added $A = CR$, independent columns times a row-reduced factor, and argued it should be the first one taught, because rank falls out of it immediately.

The same quadratic form, three chapters running

Symmetric matrices and quadratic forms are one object seen twice, and this gallery now meets it three times. In Beltrami it is the first fundamental form, whose intrinsic curvature Gauss showed is a property of the form alone. In Klein it is the absolute, whose signature selects whether you are doing Euclidean, hyperbolic or elliptic geometry. And under this series’ own hex grid it is

$Q(q,r) \;=\; q^{2} + qr + r^{2} \;=\; \begin{pmatrix} q & r\end{pmatrix}\begin{pmatrix} 1 & \tfrac12 \\[2pt] \tfrac12 & 1\end{pmatrix}\begin{pmatrix} q \\ r\end{pmatrix}$

with eigenvalues $\tfrac32$ and $\tfrac12$, determinant $\tfrac34$, both eigenvalues positive, so the form is positive definite — a Strang-style check that takes one line and settles it. That is the Eisenstein norm form, discriminant $-3$, and the six hex neighbours are its vectors of norm one. `Orthogenesis/Geometry/HexForm.lean` records this in the repository; the form and its definiteness are proved there, the eigenvalue computation is not, and the distinction matters. The same form is already at work elsewhere in the series under its other name: Book 4 · The Closing Field writes it as the Eisenstein norm $T = m^{2}+mn+n^{2} = N_{\mathbb{Q}(\omega)/\mathbb{Q}}(m+n\omega)$ and uses it as the Caspar–Klug T-number classifying which hexagonal shells close. A symmetric matrix, a lattice, a norm form and a virus capsid are four readings of one object.

Place in the Series

The honest register for a Strang citation is pedagogical rather than foundational. His books trade rigour for the big picture deliberately, and that trade is the right one for a first course and the wrong one for a proof obligation — a distinction this corpus has reason to keep sharp. Where a result needs the spectral theorem as a hypothesis discharged rather than as a fact recalled, the citation should be to the statement and its conditions, not to the course that made it memorable. Mathlib has the finite-dimensional spectral theorem for self-adjoint operators, so unlike most of the machinery this series reaches for, this one can be put under the kernel today.

References