Linear Algebra for AI and Machine Learning: A Visual and Conceptual Guide to Vectors, Matrices, and Eigenvalues

Linear algebra is usually taught twice, and the two courses never meet. Once as procedures without pictures — row-reduce this, take that determinant — and once, years later, as the shapes inside a machine-learning model without the theory. This book is written for the place where they should have met.It is built on a single idea: a matrix is a geometric action on space. It rotates, stretches, shears, reflects and flattens, and every theorem in linear algebra describes that action. The singular value decomposition says that every matrix, of any shape, is a rotation, a stretch along perpendicular axes, and another rotation. Rank, determinant, condition number, low-rank approximation and principal components can all be read off one picture: the unit circle and the ellipse a matrix turns it into.What the book contains• Twenty chapters in five parts: vectors; matrices as maps; solving and fitting; eigenvalues and the SVD; and linear algebra inside learning machines.• Ninety-one figures that carry the argument — grids being sheared, circles becoming ellipses, projections, subspaces, spectra — so that every idea is seen before it is computed.• Every theorem proved, with short proofs: rank–nullity, the four fundamental subspaces, the spectral theorem, the singular value decomposition, Eckart–Young, and the convergence of gradient descent.• Eighty-five worked examples on 2×2 and 3×3 matrices, small enough to check with a pencil.• One hundred and sixty exercises with answers, sixteen problems that cross the chapters, a summary of key ideas for every chapter, and a glossary.Connected to modern machine learningCosine similarity and attention as dot products. Embeddings as points in a space of possibilities, and why high-dimensional spaces hold far more nearly perpendicular directions than they have coordinates. Backpropagation as a product of Jacobians taken in the right order. Least squares, ridge regression and PCA as projections. Low-rank adapters for fine-tuning large models, explained by Eckart–Young. Orthogonalised optimisers as steepest descent in the spectral norm. The attention mechanism as a low-rank bilinear form.Who it is forMathematics, engineering and data-science students, and anyone who uses machine learning and wants to understand the mathematics underneath it. Comfort with algebra and a first course in calculus are enough; no previous linear algebra is assumed.The mathematics under the most capable learning systems yet built is short, geometric and beautiful. A reader who can see what a matrix does to space understands their working parts. This book shows you how to see it.
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