Linear algebra has a reputation for being abstract, and eventually it earns it -- but almost none of that abstraction is new arithmetic. It's a new way of organizing arithmetic you already know how to do. Before this course asks you to think about vector spaces, subspaces, and transformations, it needs to be certain the raw materials are solid: what "linear" actually means, how a point becomes a pair of coordinates and a coordinate pair becomes a distance, and how the trig identities you met long ago collapse into the one fact -- the cosine of an angle between two directions -- that this course will use over and over. None of that is new content. It's the toolbox getting checked before the real building starts.
This chapter also introduces two habits of mind that matter more here than they did in earlier courses: sigma notation and functions read as mappings. Sigma notation isn't decoration -- it's the bookkeeping that lets an equation with three unknowns and an equation with three hundred unknowns look like the same equation, which is exactly the compression linear algebra depends on. And thinking of a function as an assignment from one set (its domain) to another (its codomain), rather than just a formula you plug numbers into, is the same lens you'll soon point at matrices themselves -- a matrix, it turns out, is nothing more than a function that happens to eat and produce vectors.
The back half of the chapter turns from vocabulary to computation you've already brushed against. Solving a system of two equations gets done three separate ways -- substitution, elimination, and a picture of two lines crossing -- because this course will spend its first real chapter generalizing every one of those three ways to systems far too large to picture. Matrices, the row operations that simplify them, and the determinant of a 2x2 matrix all make an appearance too, not as new material but as a check that the mechanical skills a full course will soon rely on are already automatic.
The chapter closes somewhere unexpected for a review chapter: reading and writing a proof. Linear algebra is the first course in most students' path where a claim like "if a set of vectors satisfies this property, then it satisfies that one" needs to be established, not just computed. You won't be asked to produce research-grade proofs here, but you will be asked to follow an argument line by line and to write short ones of your own -- so this chapter ends by making sure "if... then," "for all," and "suppose not" are already comfortable words before a theorem asks you to use them.
Nothing in these eight sections should feel like new mathematics. If something here does feel new, that's the signal to slow down now -- because every chapter that follows will assume this one is already second nature.