Linear Algebra
The Geometry of Data
Build geometric intuition for vectors, projections, eigenvectors, the SVD, and PCA—the language used to represent and understand data.
Open textbook · Version 1.0
Mathematical Pathways
to Machine Learning
Calculus, Linear Algebra, and Optimization
A clear, integrated introduction for first-year and early sophomore students—connecting the mathematics they learn with the machine learning it powers.
DOI 10.5281/zenodo.21752763
A 15-week pathway
One mathematical story,
told in three parts.
Instead of treating linear algebra, calculus, and optimization as separate courses, the book shows how they work together inside modern learning methods.
Vector Calculus
Navigating the Loss Landscape
Move from derivatives to gradients, Jacobians, Hessians, Taylor models, and backpropagation for functions of many variables.
Optimization
The Algorithmic Journey
Turn mathematical structure into algorithms through gradient descent, Newton’s method, KKT conditions, regularization, and SVMs.
Built for learning
From the page
to working intuition.
Each topic moves through a deliberate rhythm: understand the mathematics, see its role in machine learning, and test it with a small computation.
DERIVE
See the mathematics
Definitions, geometric explanations, and detailed hand calculations develop the ideas from first principles.
CONNECT
Meet the ML model
PageRank, PCA, regression, backpropagation, regularization, and SVMs show where each tool is used.
COMPUTE
Rebuild it in NumPy
Short programming exercises translate the calculations into transparent numerical experiments.
Where to begin
Choose the path
that fits your goal.
The book assumes one semester of calculus and introductory programming, but no prior course in linear algebra, multivariable calculus, or optimization.
01
Minimal track
Definitions, key ideas, and one worked example per section.
02
Standard course
All worked examples, exercises, and exam-practice problems.
03
ML preparation
Add every ML link and reproduce the numerical work in NumPy.
Permanent scholarly record
Read it. Use it.
Cite the archived edition.
Zenodo preserves the citable Version 1.0 record. Corrections and future editions will be released as explicitly numbered versions.
Xiantao Li, Mathematical Pathways to Machine Learning: Calculus, Linear Algebra, and Optimization, Version 1.0, Zenodo, 2026. https://doi.org/10.5281/zenodo.21752763