IA y Machine Learning
Machine learning, generación aumentada por recuperación (RAG), aplicaciones de IA y web crawling.
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
Leer artículo →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
Leer artículo →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),
Leer artículo →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
Leer artículo →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
Leer artículo →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
Leer artículo →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
Leer artículo →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
Leer artículo →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
Leer artículo →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
Leer artículo →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
Leer artículo →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
Leer artículo →Notation and Formula Reference
A compact reference for the symbols and identities used throughout the book.
Leer artículo →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
Leer artículo →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
Leer artículo →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
Leer artículo →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
Leer artículo →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
Leer artículo →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
Leer artículo →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
Leer artículo →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
Leer artículo →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
Leer artículo →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
Leer artículo →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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