Determinants
The determinant compresses an entire square matrix into a single number that answers two questions at once: is this matrix invertible? and by how much does it scale volume? It is less central to computation than students
閱讀文章 →Vector Spaces and the Four Fundamental Subspaces
This is the conceptual heart of linear algebra. Everything so far — vectors, systems, matrices — now gets organized by a single powerful abstraction: the vector space and its subspaces. The payoff is the Fundamental Theo
閱讀文章 →Linear Transformations and Change of Basis
We have treated matrices as arrays of numbers. This chapter elevates them to their true identity: matrices are linear transformations once a basis is fixed. This view explains why matrix multiplication is defined the way
閱讀文章 →Orthogonality, Projections, and Least Squares
Orthogonality is where linear algebra becomes a tool for data. When an exact solution to = does not exist — the everyday situation with real, noisy, over-determined data — orthogonality tells us how to find the best appr
閱讀文章 →Eigenvalues and Eigenvectors
Eigenvalues reveal the hidden axes of a transformation — the special directions a matrix merely stretches without rotating. They explain the long-term behavior of dynamical systems, the principal directions of data, the
閱讀文章 →Symmetric Matrices, the Spectral Theorem, and Quadratic Forms
Symmetric matrices ( = ) are the best-behaved objects in linear algebra, and — not coincidentally — the ones that appear most in machine learning: covariance matrices, Gram matrices, kernel matrices, and Hessians are all
閱讀文章 →The Singular Value Decomposition
If one theorem deserves the title “fundamental theorem of data science,” it is the singular value decomposition. The SVD factors any matrix — square or rectangular, full-rank or not — into a rotation, a scaling, and anot
閱讀文章 →Matrix Decompositions, Norms, and Numerical Linear Algebra
On a real computer, linear algebra is done in finite-precision arithmetic, where the order of operations and the conditioning of a problem decide how many correct digits survive. This chapter consolidates the major matri
閱讀文章 →Tensors and Matrix Calculus
Two ideas carry linear algebra the last mile into deep learning. First, tensors generalize vectors and matrices to higher-order arrays — the natural container for images, sequences, and batches. Second, matrix calculus e
閱讀文章 →Linear Algebra in Machine Learning, Deep Learning, and AI
This final chapter is the payoff. Every concept in the book reappears here as the working machinery of modern AI. The thesis is simple and, by now, earned: linear algebra is the language in which machine learning is writ
閱讀文章 →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 weigh
閱讀文章 →Why this phase matters
Almost everyone who “bounces off” machine learning does so for the same reason: they tried to learn the algorithms before they had the language the algorithms are written in. That language is three branches of mathematic
閱讀文章 →Orientation
Classical (“tabular”) machine learning is still where most production value is created outside of the LLM frontier: churn prediction, fraud, pricing, demand forecasting, ranking. It is also the best place to build the di
閱讀文章 →From linear models to neural networks
A neural network is a stack of learned feature transformations followed by a linear model. Each layer applies an affine map then a nonlinearity; composing many of them lets the network represent functions that no single
閱讀文章 →Why images need special architectures
An image is a grid of pixels with strong spatial structure: nearby pixels are correlated, and patterns (edges, textures, objects) can appear anywhere. A plain MLP ignores this — it would need a separate weight for every
閱讀文章 →Why this is the highest-leverage phase
The Transformer is the single most important architecture in modern machine learning. It powers large language models, underpins modern vision (ViT, Phase 3), audio, code, and multimodal systems, and is where the overwhe
閱讀文章 →Discriminative vs. generative
Everything so far has been mostly discriminative: learn p(y ) to predict a label. Generative models learn the data distribution p( ) itself, so you can sample new examples that look like the training data — images, audio
閱讀文章 →A different learning paradigm
Supervised learning has labeled answers; reinforcement learning (RL) has only reward. An agent takes actions in an environment, which returns a new state and a scalar reward; the agent must learn a behavior (policy) that
閱讀文章 →Why MLOps is your edge
A model in a notebook creates zero value. Value comes from a model that serves predictions reliably, cheaply, and correctly — and keeps doing so as data shifts. This is software and systems engineering with a few ML-spec
閱讀文章 →The purpose of capstones
Everything before this phase built capability; capstones prove and compound it. A finished, deployed project that solves a real problem teaches more — and signals more — than another course completed. The strategic move
閱讀文章 →Probability & Statistics — Preface: how to use this book
If linear algebra is the language of machine learning and calculus is its engine, then probability and statistics are its worldview. Machine learning is, at its core, the discipline of drawing reliable conclusions from n
閱讀文章 →Distributions and Identities Reference
A compact reference for the notation, distributions, and identities used throughout the book.
閱讀文章 →What Is Probability?
Probability is the mathematics of uncertainty — a precise language for reasoning about things we cannot predict with certainty. A coin flip, tomorrow's weather, whether an email is spam, what word comes next in a sentenc
閱讀文章 →Conditional Probability and Bayes' Theorem
Almost all useful probability is conditional: we want to know the chance of something given what we already know. How likely is disease given a positive test? Spam given the word “free”? The next word given the sentence
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