Volume I — Matrices, Elimination, and Subspaces

This is the volume that most linear algebra courses are, done with the computer in the room.

We start where Strang starts, with the columns. A matrix times a vector is a combination of the columns of the matrix, and once that sinks in, the column space, the rank, and the reason \(Ax=b\) sometimes has no solution all follow from the same picture. §1.1 multiplies matrices four different ways — inner products, a sum of outer products, blocks, and index notation — because each way makes a different theorem obvious, and because one of them will turn out to be how attention is written down in Volume VIII.

Then elimination. §1.2 is the exemplar notebook of the course: you write Gaussian elimination with partial pivoting yourself, you watch it produce \(L\) and \(U\), you check that \(PA = LU\) to thirteen digits, and then you find the matrix where leaving the pivoting out costs eight digits of the answer. Everything after that — the inverse we refuse to form, the rank we cannot quite pin down, the four subspaces, the determinant that is conceptually central and computationally almost useless — is elimination’s consequence.

Two notebooks in this volume are more abstract than the rest, deliberately. §1.5 asks what a vector space is when the vectors are polynomials or matrices rather than columns of numbers, and §1.6 separates a linear map from the matrix that represents it, which is the distinction the whole idea of a change of basis depends on. The payoff is not aesthetic. Half of Volume III is the search for a basis in which a matrix looks simple, and you cannot search for something you have no word for.

One warning, which §1.3 makes concrete and Volume IV finally resolves. In exact arithmetic the rank of a matrix is a number. In floating point it is a decision. We will keep running into that, and it is not a defect of the machine.