Part III — The Four Fundamental Subspaces

Every transformation $A$ does two things at once: it reaches certain places and it destroys certain information. Picture a matrix that flattens 3D space onto a tilted plane. The plane it lands on is everything the transformation can produce — and the directions that got crushed to nothing are everything it loses. Part III gives these intuitions exact names, exact dimensions, and an organizing diagram so powerful that Gilbert Strang built an entire pedagogy around it: the four fundamental subspaces.

The big question here is the one that secretly drives all of applied linear algebra: given the equation $A\mathbf{x} = \mathbf{b}$, which right-hand sides $\mathbf{b}$ are reachable, and when a solution exists, how many are there? The answer splits cleanly into four subspaces — two living in the input space and two in the output space — and they are not independent. They lock together through a single relationship, rank–nullity, that says the dimensions must balance: what the transformation reaches plus what it destroys always adds up to the dimension you started with.

Chapter 13, Column Space and Null Space, introduces the first two players. The column space $C(A)$ is the set of all reachable outputs — exactly the $\mathbf{b}$'s for which $A\mathbf{x} = \mathbf{b}$ has a solution — while the null space $N(A)$ is the set of inputs that $A$ sends to zero, the "destroyed" directions that explain why solutions, when they exist, come in infinite families. Chapter 14, Row Space, Left Null Space, and Rank–Nullity, completes Strang's picture with the row space $C(A^{\mathsf{T}})$ and the left null space $N(A^{\mathsf{T}})$, then proves the rank–nullity theorem that ties all four together. The single number $\operatorname{rank}(A)$ — the true dimension of what the transformation can produce — turns out to govern everything.

Chapter 15, Dimension, Basis, and Coordinates, steps back to ask a foundational question: how many numbers do you actually need to name a vector? That count is the dimension, a basis is a minimal set of building-block directions, and coordinates are the recipe for writing any vector in terms of that basis. Chapter 16, Change of Basis, then makes the book's central theme vivid: the same vector and the same transformation can be written with completely different numbers depending on the coordinate system you choose. The matrix changes; the transformation does not. This is your first real encounter with similarity, and the visualizer returns here to show a transformation re-gridded into new coordinates — a preview of the diagonalization that will dominate Part V.

Part III closes with Chapter 17, Application: Linear Regression, where the abstraction suddenly earns its keep in spectacular fashion. Fitting the best line (or plane, or hyperplane) to noisy data is exactly the problem of finding the point in the column space $C(A)$ closest to your data vector $\mathbf{b}$ — least squares is projection onto a subspace. The entire machinery of data fitting, used daily across statistics, economics, and machine learning, is the four-subspaces picture in action.

Two recurring themes peak in this part. First, the four fundamental subspaces are the organizing framework for all of linear algebra — every later topic, from orthogonality to the SVD, connects back to this diagram. Second, geometry and algebra are one object: the rank of a matrix is at once an arithmetic count of pivots, a geometric statement about the dimension of an image, and the answer to a data-science question about how much information a dataset really contains.

This is also where the abstraction of Chapters 5 and 6 finally cashes out. Dimension, basis, and subspace stopped being definitions to memorize and became the precise language for saying what a transformation can and cannot do. If those earlier ideas felt unmotivated, Part III is the payoff — keep a concrete small matrix in front of you and watch each subspace appear in its row reduction.

By the end of Part III you will be able to: find bases for all four fundamental subspaces of a matrix directly from its row-reduced form; state and apply the rank–nullity theorem to predict the number of solutions to $A\mathbf{x}=\mathbf{b}$; compute the dimension of a space and express vectors in any chosen basis; perform a change of basis and explain why it leaves the underlying transformation unchanged; and set up and interpret linear regression as orthogonal projection onto a column space.

That last idea — the closest point in a subspace — is so important it deserves its own part. Part IV is all about orthogonality: right angles, projections, and the geometry of "closest," made precise enough to power least squares, QR factorization, Fourier analysis, and the rotations of quantum mechanics.

Chapters in This Part