AI & Machine Learning
Machine learning, retrieval-augmented generation (RAG), AI applications and web crawling tutorials.
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
Read article →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 chain rule — is the ma
Read article →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 (optimization), approximate complicated functions by simple ones (linearization and Newton's method),
Read article →The Integral and the Fundamental Theorem
Integration is the second pillar of calculus: the mathematics of accumulation. Where the derivative chops a curve into instantaneous rates, the integral sums infinitely many tiny pieces into a total — an area, a distance
Read article →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
Read article →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 im
Read article →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
Read article →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 gradient — the v
Read article →Multiple Integrals and a Glimpse of Vector Calculus
Just as differentiation extended to many variables, so does integration. Multiple integrals accumulate a quantity over a region of two, three, or more dimensions — the natural setting for probabilities over several varia
Read article →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
Read article →Calculus in Machine Learning, Deep Learning, and AI
Here is the payoff. Linear algebra gives AI its vocabulary — vectors, matrices, transformations. Calculus gives it the ability to learn. At the heart of almost every modern machine-learning system is one computational pa
Read article →Calculus — Preface: how to use this book
If linear algebra is the language of machine learning, calculus is its engine of change. Calculus is the mathematics of things that vary — how a quantity responds to a small nudge in another (the derivative), and how cou
Read article →Notation and Formula Reference
A compact reference for the symbols and identities used throughout the book.
Read article →Vectors and the Geometry of Space
Everything in linear algebra is built from one object: the vector. Before we can multiply matrices or diagonalize transformations, we need to be completely comfortable with what a vector is, how to combine vectors, and h
Read article →Systems of Linear Equations
A system of linear equations 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
Read article →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
Read article →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
Read article →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
Read article →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
Read article →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
Read article →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
Read article →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
Read article →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
Read article →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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