Chapter 1: Systems of Linear Equations: The Geometry of Planes

Linear algebra begins with a plain question: when do several linear conditions hold at once? Each single equation in three unknowns, such as $x+y-z=0$, is easy on its own — its solutions form a flat plane floating in space. The interesting object is the system: two or three such equations demanded simultaneously. Algebraically that is a stack of equations; geometrically it is a stack of planes, and the solutions of the system are exactly the points where all of the planes pass through one another.

This chapter works in ${ℝ}^3$ on purpose. Three dimensions is the largest space we can actually see, and it is big enough to show every behavior a linear system can have: a unique solution, a whole line or plane of solutions, or no solution at all. Instead of memorizing those cases from a table, you will orbit them. You will drag a plane through space and watch the intersection line it carries slide with it, push three planes into and out of agreement, and see why the elimination moves you learned in algebra never disturb the set of solutions.

By the end of the chapter, "solve the system" should feel like a geometric act: find the shape that the planes cut out together. Everything later in the book — spans, transformations, eigenvectors, least squares — refines this one picture.