All Visualizations

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

(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

Any single category of a multinomial is Binomial, and merging categories keeps it Multinomial (B&H 7.4.4) — a marginal is just lumping everything else into "other". Empirical draws land on the exact Binomial pmf

Covariance as averaged co-deviation, its unit-dependence flaw, and correlation as the scale-free fix in [−1,1] — plus the two traps: it only sees straight lines, and uncorrelated ≠ independent (Y = X²)

A normal random vector as a cloud in the plane: every projection is a bell, μ and Σ pin it down, Σ's eigenvectors are the ellipse axes, uncorrelated ⇒ independent (the Gaussian miracle), and conditioning gives the regression line

M(t)=E[e^{tX}] as one function that packages a distribution: the Taylor expansion showing moments ARE the coefficients, reading them off as derivatives at 0 (M′(0)=E[X], Var=M″(0)−M′(0)²), sums becoming products (n exponentials → (λ/(λ−t))ⁿ = Gamma's MGF), and uniqueness as the fingerprint that makes it a proof

How a density transforms under Y = g(X): conserving probability gives f_Y = f_X/|g′(x)| in 1D; in 2D the Jacobian is the local linear map, |det J| is the area-scaling factor, and f_Y = f_X/|det J| — visualized with grid deformation, a |det J| field, and a pushed-through Gaussian cloud

The density of T = X + Y (B&H Thm 8.2.1), two ways: summing the independent joint PMF along the anti-diagonal x+y=t, and the continuous flip-and-slide — f_T(t) = ∫ f_X(x)f_Y(t−x)dx as the overlap area of f_X with a flipped, shifted f_Y. Shows uniform⊛uniform = triangle and why sums smooth toward a bell

A full worked example: X ~ Uniform(0,a) plus Y ~ Exp(λ). Flip-and-slide the exponential, watch the integral's upper limit switch from t to a, and derive the piecewise density — (1/a)(1−e^{−λt}) rising to a peak at t=a, then (1/a)(e^{λa}−1)e^{−λt} decaying — a smoothed exponential

The distribution of a probability: two shape parameters sculpt every shape on [0,1] (Uniform, U, skew, spike), the mean α/(α+β) and concentration α+β, and the conjugate-prior magic — flip coins and watch Beta(α,β) → Beta(α+s, β+f) update by just adding counts, with α,β as pseudo-counts

The waiting time for the α-th event: shape α and rate λ, the Exponential as α=1, Gamma as a sum of α exponential waits (time to the α-th Poisson event) drifting to a bell as α grows (CLT), mean α/λ and variance α/λ², and the Beta–Gamma split X/(X+Y) ~ Beta

Discrete-time, discrete-state chains via a 3-state weather chain: the memoryless Markov property, the transition matrix (rows are conditional distributions summing to 1), a distribution evolving by π ↦ πP, the stationary distribution πP=π that every start converges to, why convergence needs irreducibility + aperiodicity (a period-2 chain oscillates forever), and reversibility / detailed balance as the engine behind MCMC

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

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

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

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

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

Why composing linear layers collapses to a single line (w₂(w₁x+b₁)+b₂ is still affine), so a nonlinearity is essential; ReLU as the simplest bend, one unit as a placeable hinge, and summing hinges to trace any curve — plus why its gradient of 1 beats sigmoid/tanh saturation

Prince Fig 8.4 — three ReLU units with frozen weights and joints pinned at 1/3 and 2/3. Four parameters span exactly the 3-segment piecewise-linear functions, and nothing else — which is bias, made visible

Volume flees to the corners, everything lives in the shell, and distances concentrate until "nearest" stops meaning anything — then why deep learning survives 150,528 dimensions anyway: intrinsic, not ambient, dimension is what bites

Infinitely many functions fit the same points exactly, so the data cannot choose — something else must. Five interpolants, all with zero training error, and the no-free-lunch punchline: the assumption that wins on a sine is catastrophic on a step

Past the interpolation threshold, training error stays pinned at zero while the fitted curve smooths out and ‖w‖ collapses — why hugely overparameterised networks generalise. Every curve computed, none illustrated

The three sources of test error, isolated one at a time on a polynomial-fit playground — the irreducible noise floor, a model too simple, a model too twitchy — then recombined into the U-shaped total

Logits → softmax → probabilities → L = −log(p_c), and the gradient that falls out: ∂L/∂z = p − y. Shows why squared error stalls exactly when the model is confidently wrong

A fine-grained computation graph for one sigmoid neuron — forward pass node-by-node, then the chain rule flowing backward via each node's local rule (+ copies, × swaps, σ scales by a(1−a))

A 3×3 kernel slides and takes a dot product to build a feature map; kernels as learned edge/blur detectors, parameter sharing → translation equivariance (shift the input, the map shifts), stride/padding sizing, max-pooling, and the receptive field growing with depth

The depth axis untangled: an image as an H×W×C stack, one filter spanning all input channels to collapse them into a single feature map, K filters making K output channels, why space is weight-shared but channels are dense, channels as feature types, and the 1×1 convolution as pure channel mixing

The "not four copies" lesson: one filter answers one question, why running the same filter four times would be redundant, four different learned filters giving four different feature maps, output channels as four different questions, untangling the within-filter (input-channel) index from the across-filter index in h^(k), and K as a design dial you drag

The transformer's core op, step by step: tokens as vectors, the shared W_Q/W_K/W_V projections (parameter sharing across positions), scores as dot products q_i·k_j, softmax into attention weights, and the output as a weighted sum of values — closing on the shared-weights-vs-dynamic-attention distinction

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