The mathematical foundations of data science, built from the ground up with Python. Covers sets through graph algorithms — including calculus, linear algebra, and multivariable optimization — with every concept grounded in real ML and data science applications. Implement gradient descent from scratch, visualize Riemann sums converging to integrals, manipulate linear transformations in real time, and run Gram-Schmidt orthogonalization step by step. 28 lessons, 56 graded exercises, and 4 interactive simulators designed for engineering undergraduates.
Sets and Set Operations for Data Science
Relations and Functions: The Language of Mappings
What is the **cardinality of the power set** of a set with n elements, and why?
A function f: A → B is a **bijection** when it is:
Lines and Linear Models
Quadratic Functions: Curves, Vertices, and Optimization
Two lines are **parallel** if they have the same slope, and **perpendicular** if...
Polynomials: Arithmetic, Roots, and Behavior
Composite and Inverse Functions
A polynomial of degree n can have at most how many real roots and turning points...
Exponential Functions: Growth, Decay, and the Natural Base
Logarithms: Properties, Applications, and Information Theory
The three fundamental logarithm rules are: log(a·b) = log(a) + log(b), log(a/b) ...
Pre-Calculus Synthesis
Sequences and Convergence
Limits and Continuity
The famous limit (1 + 1/n)ⁿ as n → ∞ converges to:
Derivatives: Rules and Computation
Critical Points and Optimization
What does the **chain rule** state, and why is it critical for training neural n...
Riemann Sums and the Definite Integral
Interactive: Riemann Sum Visualizer
Integration Applications: Area, Probability, and Economics
The Fundamental Theorem of Calculus states that if F'(x) = f(x), then ∫ₐᵇ f(x)dx...
Vectors, Matrices, and Determinants
Solving Linear Systems
What does a **zero determinant** of a square matrix A tell you?
Linear Algebra Foundations Lab
Vector Spaces, Span, and Linear Independence
Basis, Rank, Null Space, and the Rank-Nullity Theorem
The Rank-Nullity Theorem states that for an m×n matrix A: rank(A) + nullity(A) =...
Linear Transformations: Mappings That Preserve Structure
Interactive: Linear Transformation Visualizer
Kernel and Image of a Linear Transformation
What are the **kernel** and **image** of a linear transformation T, and how do t...
Norms, Inner Products, and Distance in Feature Space
Orthogonality and the Gram-Schmidt Process
Interactive: Gram-Schmidt Visualizer
Which of the following are true about orthonormal vectors? (Select all)
Partial Derivatives and the Gradient Vector
Gradient Descent: Optimizing Functions Computationally
Interactive: Gradient Descent Visualizer
The Hessian Matrix and Classifying Critical Points
When the Hessian determinant is negative at a critical point, the point is class...
Optimization in Machine Learning
Graph Representation, BFS, DFS, and Topological Sort
Shortest Path Algorithms
Minimum Spanning Trees and Graph Reachability
Which shortest-path algorithm handles negative edge weights (without negative cy...
Graph Algorithms in Practice
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