Part I — Vectors and Systems

Hold up your hand and push it through the air. Your hand sweeps out a path, and every point of that motion can be described by a few numbers and a few rules: how far, in which direction, combined how. That is the whole secret of linear algebra in one gesture — space, and the ways we can move through it. Before we touch a single matrix, you need the raw materials of that motion: the vector, the arrow that is a direction-and-magnitude, and the linear system, the set of constraints that pins down where arrows can land. This part builds both from the ground up, picture first, formula second.

The big question Part I answers is deceptively simple: what does it mean to combine things linearly, and what can you build out of those combinations? The answer turns out to organize everything that follows. A weighted sum of vectors — scale a few arrows, add them tip to tail — is the single operation linear algebra is about. Solve a system of equations, fit a line to data, render a 3D scene, run a search engine: under the hood, you are always asking which linear combinations are reachable and which are not.

Chapter 1, What Is Linear Algebra?, opens with the transformation viewpoint that threads through the entire book — the idea that a matrix is a function that transforms space — and introduces the recurring 2D visualizer you will see in nearly every geometric chapter from here on. Chapter 2, Vectors, makes the arrow concrete: addition and scalar multiplication as geometry, then as lists of components $\mathbf{v} = (v_1, v_2, \dots, v_n)$, and the realization that an arrow and a list of numbers are the same object seen two ways. Chapter 3, Systems of Linear Equations, asks what a collection of linear constraints looks like — intersecting lines and planes — and why a system has no solution, exactly one, or infinitely many; here you'll meet PageRank for the first time, informally, as a system waiting to be solved. Chapter 4, Gaussian Elimination and Row Reduction, turns that geometry into a reliable procedure: elementary row operations, echelon and reduced row-echelon form, pivots and free variables — the workhorse algorithm you'll implement from scratch in your toolkit.

Then the ground tilts upward. Chapter 5, Vector Spaces, asks you to let go of arrows-in-the-plane and accept that polynomials, functions, and even matrices are vectors too, because they obey the same eight axioms. This is the first genuinely abstract climb in the book, and it is honestly a little vertiginous — the payoff is that everything you learn once now applies everywhere at once. Chapter 6, Subspaces, Span, and Linear Independence, brings the abstraction back to earth: the span of a set of vectors is just all the linear combinations you can reach, independence means no vector is redundant, and together they aim you straight at the idea of a basis — the minimal set of arrows that builds an entire space.

Two themes from the style bible begin here and never leave. First, geometry and algebra are two views of one object: solving a system is finding where surfaces intersect; a span is a line, plane, or hyperplane through the origin. Second, computation validates theory — every hand calculation in this part is checked against a short numpy snippet, so you build the algorithm by hand and then watch the machine confirm it.

A word on difficulty. Chapters 1 through 4 are gentle and concrete; you can almost see the answers. Chapters 5 and 6 are where many students first feel linear algebra get abstract, and that feeling is normal, not a sign you're lost. The trick is to keep a concrete example in mind — $\mathbb{R}^2$, or the plane, or a small system — and check every axiom and definition against it. Abstraction in this book is never decoration; it always earns its keep by unifying things you already understand.

By the end of Part I you will be able to: add and scale vectors and read them as both arrows and coordinates; set up a system of linear equations and predict the shape of its solution set before solving it; run Gaussian elimination to reduced row-echelon form by hand and explain what each pivot and free variable means; state the vector-space axioms and recognize non-obvious examples (polynomials, functions, matrices) as vector spaces; and decide whether a set of vectors is linearly independent and describe the subspace it spans. You'll also have started your from-scratch toolkit/, with working vectors.py and linear_systems.py modules verified against numpy.

That sets the stage for the book's central move. In Part II we stop treating matrices as bookkeeping grids for systems and reveal what they really are: functions that transform space. The row-reduction machinery you just mastered becomes one tool among many, and the matrix steps into its true role as the representation of a linear map.

Chapters in This Part