The Derivative
The derivative is the central object of differential calculus and the single most important concept for machine learning. It answers the tangent problem: the instantaneous rate at which a function changes. Once we have i
記事を読む →Differentiation Rules
Computing every derivative from the limit definition would be exhausting. Fortunately, a small set of rules lets us differentiate almost any function mechanically. The most important of these — the — is the mathematical
記事を読む →Applications of the Derivative
Now that we can compute derivatives, we put them to work. Derivatives find the highest and lowest points of a function (), approximate complicated functions by simple ones ( and Newton's method), and resolve stubborn lim
記事を読む →The Integral and the Fundamental Theorem
Integration is the second pillar of calculus: the mathematics of . Where the derivative chops a curve into instantaneous rates, the integral sums infinitely many tiny pieces into a total — an area, a distance, a probabil
記事を読む →Techniques of Integration
Differentiation is mechanical; integration is an art. There is no universal algorithm, so we build a toolkit of techniques — each one reversing a differentiation rule — plus numerical methods for the (many) integrals wit
記事を読む →Applications of Integration
Integration is far more than ``area under a curve.'' Any quantity built by accumulating infinitely many infinitesimal pieces is an integral: areas between curves, volumes of solids, lengths of arcs, averages, and — most
記事を読む →Sequences, Series, and Taylor Series
Can you add up infinitely many numbers and get a finite total? Sometimes — and when you can, the result is one of the most powerful tools in mathematics: the ability to represent complicated functions as infinite polynom
記事を読む →Multivariable Calculus: Partial Derivatives and the Gradient
Machine-learning models depend on millions of parameters at once, so single-variable calculus is not enough. This chapter extends differentiation to functions of several variables. The star object is the — the vector of
記事を読む →Multiple Integrals and a Glimpse of Vector Calculus
Just as differentiation extended to many variables, so does integration. accumulate a quantity over a region of two, three, or more dimensions — the natural setting for probabilities over several variables. The formula i
記事を読む →Optimization and Differential Equations
This chapter gathers the calculus that most directly becomes machine learning: how to find minima of functions of many variables (with and without constraints), the gradient-descent dynamics that training uses, and the d
記事を読む →Calculus in Machine Learning, Deep Learning, and AI
Here is the payoff. Linear algebra gives AI its — vectors, matrices, transformations. Calculus gives it the ability to . At the heart of almost every modern machine-learning system is one computational pattern: compute a
記事を読む →Preface: how to use this book
If linear algebra is the language of machine learning, calculus is its . Calculus is the mathematics of things that vary — how a quantity responds to a small nudge in another (the derivative), and how countless tiny piec
記事を読む →Notation and Formula Reference
A compact reference for the symbols and identities used throughout the book.
記事を読む →Vectors and the Geometry of Space
Everything in linear algebra is built from one object: the . Before we can multiply matrices or diagonalize transformations, we need to be completely comfortable with what a vector is, how to combine vectors, and how to
記事を読む →Systems of Linear Equations
A asks: which values of the unknowns satisfy several linear constraints at once? This is the oldest problem in the subject and still its computational core — training a linear model, balancing a chemical equation, and so
記事を読む →Matrices and Matrix Algebra
A matrix is the workhorse object of linear algebra. This chapter develops the algebra of matrices — how to add, multiply, transpose, and invert them — always keeping sight of the three views from the preface: a matrix is
記事を読む →Determinants
The determinant compresses an entire square matrix into a single number that answers two questions at once: and It is less central to computation than students often think (we rarely compute large determinants), but its
記事を読む →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 and its . The payoff is the , which describes the four subspaces h
記事を読む →Linear Transformations and Change of Basis
We have treated matrices as arrays of numbers. This chapter elevates them to their true identity: matrices linear transformations once a basis is fixed. This view explains why matrix multiplication is defined the way it
記事を読む →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 approxima
記事を読む →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
記事を読む →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 s
記事を読む →The Singular Value Decomposition
If one theorem deserves the title ``fundamental theorem of data science,'' it is the singular value decomposition. The SVD factors matrix — square or rectangular, full-rank or not — into a rotation, a scaling, and anothe
記事を読む →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
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