Calculus and Linear Algebra

The maths under graphics, machine learning and simulation, computed in Python: limits, derivatives, optimization, integrals and series, then vectors, matrices, linear systems, determinants, transformations and eigenvalues.

1
Functions and limits
What a limit is, how to find one by getting closer and closer, and the two famous limits: sin(x)/x and (1 + 1/n)ⁿ.
2
The derivative
The derivative as the slope of a curve at one point: the difference quotient with a shrinking step, and why f(x) = x² has slope 2x.
3
Rules for derivatives
The power, sum, product and chain rules, a table of common derivatives, and checking every rule numerically with a central difference.
4
Maxima, minima and gradient descent
Finding the best value: setting the derivative to zero (the largest field for a fence), the second-derivative test, and gradient descent when algebra is too hard.
5
Integrals: area under a curve
Adding up thin rectangles (Riemann sums), how the error shrinks with more strips, and the fundamental theorem that links integrals to derivatives.
6
Sequences and series
Adding infinitely many numbers: geometric series that converge, the harmonic series that does not, and the Taylor series that computes e.
7
Vectors and the dot product
Vectors as arrows and as lists of numbers: adding, scaling, length, the dot product, the angle between two vectors and cosine similarity.
8
Matrices and matrix multiplication
Matrices as grids of numbers, the row-times-column rule, which shapes can be multiplied, and why A·B is usually not B·A.
9
Solving systems of linear equations
Writing equations as rows of numbers, Gaussian elimination step by step with exact fractions, and the three possible outcomes.
10
Determinants and inverses
The determinant as an area scale factor, when a matrix can be undone, and the inverse of a 2×2 matrix with exact fractions.
11
Matrices as transformations
What a 2×2 matrix does to the plane: rotations, scaling and shears, reading a matrix from its columns, and combining moves by multiplying.
12
Eigenvalues and eigenvectors
Directions a matrix only stretches: A·v = λv, the characteristic equation for 2×2 matrices, power iteration, and where eigenvalues appear in practice.