Learning Visualizations

PMF/PDF and CDF of named distributions with live parameter sliders, point and inverse-CDF queries, and moments — Normal, Binomial, Poisson, t, χ², Gamma, Beta and more

One- and two-sample t-tests, paired t-test, and chi-square goodness-of-fit — paste data, pick α, read the statistic, df, p-value and verdict

Dependence, the parallelogram test, span collapsing to a line vs filling the plane, coordinates in a non-standard basis

Col(A) as reachable outputs, rank-1 collapse, Row(A) in the input space, Null(A) ⊥ Row(A), rank theorem and rank-nullity

A transformation as a map: range as everything T can output, kernel as inputs that vanish, fibers as kernel cosets, injective ⟺ ker = {0}, surjective ⟺ range = W, and rank-nullity

Computing rank and nullity by hand: row-reduce, count pivots vs free columns, and read off the kernel basis — worked on a 3×3, a wide and a tall matrix, plus differentiation on polynomials

The row rule, column picture, rotation matrix, how A transforms the whole plane, and linearity A(u+v) = Au + Av

Column picture: Ax as a weighted sum of columns. Row picture: each output component as a dot product — rows as questions, null space as the blind spot

Three axioms, the subspaces of ℝ², closure under addition and scaling, span

Vectors, angles, and projection

Three axioms that generalise the dot product to any vector space — weighted inner products, L² function spaces, induced norm, Cauchy-Schwarz, and generalised projection

Turning any independent set into an orthonormal basis — the projection subtraction trick, why orthogonality is guaranteed, and the QR decomposition

QᵀQ = I means transpose is inverse — rotations (det = +1), reflections (det = −1), preserving lengths and angles, and frame change T′ = QTQᵀ

T(u+v) = T(u)+T(v) in any vector space — differentiation on polynomials, integration, transpose on matrices, kernel, image, rank-nullity, and matrix representation

Perpendicular vectors and parallelogram area

Torque as a cross product: M = r × F

Area scaling, orientation, and collapse — the geometry behind ad − bc

Why the diagonal sum is basis-independent — eigenvalue sum, infinitesimal determinant, and the cyclic property tr(AB) = tr(BA)

A⁻¹ undoes A — the round-trip identity and why singular matrices have no inverse

Av = λv — the special directions a matrix only stretches, never rotates

E_λ = Null(A − λI) as a subspace — geometric vs algebraic multiplicity, repeated eigenvalues, and defective (non-diagonalisable) matrices

Rank, transformation law T′ = RTRᵀ, index notation, and principal axes as eigenvectors

Why ω is orthogonal, and v = ω × r

Radial and transverse components in polar coordinates

The point about which the body is rotating instantaneously

ψ, θ, φ and transformation of displacements

r × v, equal areas, and Kepler's second law

v = rω, centripetal and tangential acceleration

Orbits are conic sections — derived from Newton's inverse-square law via Binet's substitution

Equal areas in equal times — conservation of angular momentum h = r²θ̇

T² = 4π²a³/K — period depends only on semi-major axis, not eccentricity

Visualisation of impulse and momentum

Radial & transverse equations, Binet's substitution, orbit as a conic

CoM reduction, relative coordinate, and reduced mass

χ, eccentricity from v₀ and β₀, and the Fig 4.9-2 chart

Cotangential transfer between coplanar circular orbits, Δv₁, Δv₂, transfer time

F = +K/r² produces the repulsive branch of a hyperbola — Rutherford scattering geometry

Δv at θ* = 150° transfers between two orbits sharing the same apse line — velocity vector decomposition into Δvₜ and Δvₙ

Given χ = r₀v₀²/K, β₀, and r₀/R at engine cutoff, derive e, a/R, and initial position θ₀ relative to perigee

Eccentricity, focus-directrix definition, and the polar equation

Semi-axes, area = πab, focal properties, and the bridge to orbital mechanics

Slope fields, solution curves, initial conditions, order and linearity

SHM, phase portrait, separatrix, and nonlinear period corrections

∂u/∂t = α ∂²u/∂x² — Fourier modes, diffusion, decay rates

(A∪B)ᶜ = Aᶜ∩Bᶜ and (A∩B)ᶜ = Aᶜ∪Bᶜ — visualised as Venn diagrams

P(A|B) = P(A∩B)/P(B) as shrinking the sample space — and why P(A|B) ≠ P(B|A)

Partitioning P(A), inverting conditionals, the rare-disease test, and the chain rule for n events

P(A∩B)=P(A)P(B), why independent ≠ disjoint, and conditional independence

Why every pair being independent doesn't make events mutually independent — the two-coin parity counterexample

Weighted sum, activation, decision boundary, and the learning rule

How ReLU hidden units fold the input space into a piecewise linear surface — Figure 3.8 from Prince

The motion of the instantaneous rotation axis in body and space frames

Velocity of a point on a rigid body → h_i = r_i × m_i v_i → H = Iω

Inertia tensor of a slender bar at angle θ — moments, products, and H = [I]·ω when spinning about z