Chapter 2: Vectors, Spans, and Solution Sets

Chapter 1 treated a system of equations as a stack of planes and asked where they intersect. This chapter turns the camera around. Instead of asking which points satisfy given equations, we start from vectors and ask what they can build: the set of all their scaled sums, called the span. A single vector spans a line; two vectors usually span a plane; and the word "usually" once again hides the most instructive cases.

The two viewpoints are not rivals — they are the two halves of linear algebra. The solution set of a homogeneous system turns out to be a span; the solution set of any consistent system is a span that has been picked up and translated away from the origin. And the subspaces that make all of this tick can be recognized by eye: they are the flat things through the origin, closed under the very operations — adding and scaling — that build spans in the first place.

In this chapter you will watch a span physically collapse from a plane to a line as its two spanning vectors slide into alignment, see a solution set revealed as a translated copy of its homogeneous partner, and run a visual subspace test on three candidate surfaces. The pictures are simple; the vocabulary they anchor — span, independence, homogeneous, particular, subspace — carries the rest of the book.