Mathematics for AI
The mathematical bedrock under machine learning: vectors and matrices as the geometry of data, eigenvalues and the SVD as the machinery of compression and structure, then probability and statistics as the language of uncertainty — the same topics a first-semester M.Tech AI course expects. Intuition first, a visualization you can play with, then the math.
Linear algebra
Vectors as the geometry of data, matrices as transformations, systems of equations as geometry, and the eigen/SVD decompositions that expose a matrix's structure.
- Vectors: data as arrowsA row of numbers is a point, an arrow, and a direction. The inner product is how we measure agreement. 9 min · intro
- Matrices: transformations and dataA matrix is a linear map — it moves the whole space at once — and its columns are where the basis vectors land. 12 min · core
- Linear systems: solving, and what to do when you cannotIntersecting equations, the column picture, and least squares as the principled answer to an inconsistent system. 13 min · core
- Eigenvalues and eigenvectors: the directions a matrix cannot rotateMost vectors get turned by a matrix; a special few only get stretched. Those directions are the matrix's eigenstructure. 13 min · core
- The SVD: every matrix is a rotate, a stretch, a rotateThe singular value decomposition exposes any matrix as a rotation, a diagonal scaling, and another rotation — the foundation of compression. 14 min · advanced
Probability
Probability spaces, conditional probability and Bayes, random variables and their distributions, and the inequalities that bound uncertainty.
- Probability spaces: outcomes, events, and countingA sample space, a family of events, and a rule for assigning probabilities — then every conditional quantity is just subset arithmetic. 10 min · intro
- Random variables and their distributionsA random variable assigns a number to each outcome; its distribution says how the probability is laid out across those numbers. 12 min · core
- Expectation and variance: the balance point and the spreadExpectation is the long-run average and a linear operator; variance measures spread — and Chebyshev turns that spread into a bound. 12 min · core
- Conditional probability and Bayes' rulePriors, likelihoods and evidence — how a probability about the world updates when a test comes back positive. 12 min · core
Statistics & estimation
Describing data, correlation and covariance, mean-square estimation and the orthogonality principle, and the hypothesis tests behind every result you will read.
- Describing data: location, spread, and shapeMean, median, variance, quantiles and the box plot — what each summary captures and how each responds to an outlier. 10 min · intro
- Covariance and correlation: how two variables move togetherCovariance measures joint variation, correlation rescales it into [−1, 1], and neither one implies causation. 11 min · core
- Estimation and the orthogonality principlePoint estimates, bias and mean-square error, and the projection view of least squares that makes it optimal. 13 min · advanced
- Hypothesis testing: p-values, errors, and powerA null hypothesis, a test statistic, a rejection region — and the two ways a test can be wrong. 13 min · advanced