Preface: how to use this book
Linear algebra is the mathematics of many numbers at once. The moment you stop thinking about a single quantity and start thinking about lists of quantities — the pixels of an image, the features of a customer, the weights of a neural network, the words of a document — you are doing linear algebra. It is the most useful piece of mathematics for computing, data, and artificial intelligence, and unlike calculus it is concrete: every object is something you can write down, draw, and compute.
Who this is for
This book starts from the very beginning (what a vector is) and climbs all the way to the singular value decomposition and the matrix calculus behind backpropagation. It is written for the self-learner who wants both computational fluency (“I can do the arithmetic”) and conceptual understanding (“I know what it means and why it matters”). No proof is included for its own sake; every result is motivated, and most are illustrated with a picture and a small numerical example.
The structure
Part I — Foundations (beginner): vectors, linear systems, matrices.
Part II — Core theory (intermediate): determinants, vector spaces and the four fundamental subspaces, linear transformations, orthogonality and least squares.
Part III — Advanced (expert): eigenvalues, symmetric matrices and the spectral theorem, the SVD, numerical linear algebra, and matrix calculus.
Part IV — Applications: a capstone chapter showing how every concept reappears in machine learning, deep learning, and AI.
How to read it
Read with a pen. When you meet a Definition box, restate it in your own words; when you meet an Example, cover the solution and try it first; when you meet a Try it in NumPy box, actually run the code. The colored boxes recur throughout:
Key idea — the one sentence to remember.
Intuition — the geometric or mental picture.
Common pitfall — mistakes to avoid.
How this powers ML / AI — the forward link to applications.
Try it in NumPy — a hands-on check.
Three mental models recur on every page. A matrix is simultaneously (1) a table of numbers, (2) a linear transformation that moves space, and (3) a collection of vectors (its rows or columns). Fluency means switching between these three views at will. The whole book trains that switch.