Overview

Topic 2 built an algorithm — elimination — that solves $A\mathbf{x}=\mathbf{b}$ mechanically. This topic steps back and asks the structural question the algorithm's output actually answers: what does the full solution set look like, and what invariant properties does a matrix have regardless of which basis you happen to compute in? The answer is a short, precise vocabulary — vector space, subspace, span, independence, basis, dimension — that every later topic in the unit assumes fluently. It closes on the four fundamental subspaces, the single organising picture Strang calls "the heart of linear algebra": every matrix carries exactly four subspaces attached to it, and nearly every question the rest of the unit asks (is a system solvable? is a map invertible? what are the eigenvectors?) is really a question about one of these four.