Mathematical Foundation For Machine Learning and AI

Master scalars, vectors, matrices, and tensors, and see how inputs and outputs use these structures in AI. Grasp indexing, dimensions, and key operations like addition, subtraction, multiplication, and transpose.

Explore vector and matrix norms, including L2, L1, max, and Frobenius norms, and their use in measuring magnitude, loss, regularization, and machine learning with support vector machines.

Learn to work with scalars, vectors, matrices, and tensors in Python using Jupyter notebooks, including creating arrays, indexing, shapes, dimensions, and matrix operations such as multiplication and transpose.

Explore diagonal and symmetric matrices, unit vectors, and orthogonality, and learn how normalization speeds neural network training using the euclidean norm.

Explore eigenvalues and eigenvectors, understand their theory, and apply eigen decomposition with Python for principal component analysis and data compression.

Learn to compute norms and eigenvalues with eigenvectors using the linear algebra package, perform eigen decomposition, and visualize normalization and vector scaling in a notebook.

Explore derivatives within multivariate calculus, uncovering how gradients drive neural networks through gradient descent. Learn derivative notation, product and chain rules, and partial derivatives essential for convex optimization.

Learn basics of integration, including indefinite and definite integrals, the antiderivative, the constant of integration, and area under the curve with applications in probability, convex optimization, and machine learning.

Explore vector calculus fundamentals by defining the gradient as the nabla operator of partial derivatives, showing how gradients point to the steepest ascent and guide parameter updates to minimize loss.

Visualize gradients by building a 100x100 mesh grid for f(x,y)=x*y^2, plotting a color map and gradient vectors, and verifying partial derivatives to understand gradient direction and magnitude.

Explore optimization for neural networks, including deep neural networks, by minimizing objective functions with gradients under constraints, using convex optimization and stochastic gradient descent.

Explore probability theory as the math of uncertainty and its role in AI, contrast frequentist and Bayesian approaches, and preview probability distributions used in machine learning.

Explore random variables and their probability distributions, including discrete and continuous cases, joint distributions, uniform distributions, probability mass and density functions, and marginal and conditional probabilities.

Explore the expectation, variance, and covariance of random variables, and learn how the covariance matrix summarizes their linear relationships for practical machine learning applications.

Graph probability distributions in R using a normal distribution to model IQ scores (mean 100, sd 15) and compute the region above a threshold with area under the curve.

Build a covariance matrix in R from scratch using column vectors, compute means, form differences, multiply to obtain a sample covariance, compare with built-in covariance, and extract variances.

Explore key random variables used in machine learning, including Bernoulli and multinomial for discrete data, and Gaussian, exponential, and Laplace distributions for continuous data.