Self-Study Syllabus

Organised by subject · progress reflects my actual, non-linear path

Tech tree → All visualizations → Bench track →
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Mathematical Foundations
Linear Algebra
GlossaryLinear Algebra Glossary → SourcesCatalog → Practicenone yet
3Blue1Brown · Singh Essence of Linear Algebra · Linear Algebra: Step by Step Current
Dot Product Vectors, angles, and projection 3B1B · Singh def →
vizDot Product
Cross Product Perpendicular vectors and parallelogram area 3B1B · Singh
vizCross Product vizMoment of a Force
Determinant Area scaling, orientation, and collapse — ad − bc 3B1B · Singh def →
vizDeterminant
Inverse Transformation A⁻¹ undoes A — round-trip identity, singular failure 3B1B · Singh def →
vizInverse Transformation
Eigenvectors Av = λv — directions a matrix only stretches 3B1B · Singh def →
vizEigenvectors vizTensors
Eigenspaces E_λ = Null(A − λI) — multiplicity & defective matrices 3B1B · Singh def →
vizEigenspaces
Linear Transformations Rotations, reflections, shears, projections as a unified concept 3B1B v3 def →
vizLinear Transformations vizMatrix-Vector Multiplication
Matrix Multiplication as Composition Applying A then B = BA; why order matters geometrically 3B1B v4 def →
Span, Basis & Linear Independence What a basis is; when vectors are redundant; dimension 3B1B v2 · Singh def → basis →
vizBasis, Span & Linear Independence vizVector Subspaces
Column Space & Null Space What A can reach; what it maps to zero; rank-nullity theorem 3B1B v7 · Singh def →
vizKernel & Range vizRank & Nullity vizRow Space & Column Space vizTwo Ways to Read Ax
Change of Basis Same transformation, different coordinate frame; P⁻¹AP 3B1B v13 · Singh def →
Orthogonal Projection Projecting onto a subspace; least squares Singh def →
vizGram-Schmidt vizOrthogonal Matrices vizInner Product
Eigendecomposition & Diagonalisation A = PDP⁻¹; matrix powers; why it matters 3B1B v14 · Singh def →
vizTrace
Singular Value Decomposition The most general decomposition; every matrix Singh
Probability
GlossaryProbability Glossary → SourcesCatalog → Practicenone yet
Blitzstein & Hwang — Introduction to Probability CRC Press · Harvard Stat 110 Current
Probability & Counting Sample spaces, naive definition, multiplication rule, binomial coefficients, story proofs B&H §1 def →
vizDe Morgan's Laws
Conditional Probability Bayes' rule, law of total probability, conditional independence, coherency B&H §2 def →
vizConditional Probability vizBayes' Rule
Random Variables & their Distributions PMFs, Bernoulli, Binomial, Hypergeometric, Discrete Uniform, CDFs, functions of r.v.s B&H §3 def →
toolDistribution Calculator
Expectation Linearity, indicator r.v.s and the fundamental bridge, LOTUS, variance, Geometric, Poisson B&H §4 def →
Continuous Random Variables PDFs, Uniform, universality of the Uniform, Normal, Exponential, memorylessness B&H §5 def →
Moments Interpreting moments, skewness and kurtosis, sample moments, moment generating functions B&H §6 def →
vizMoment Generating Functions
Joint Distributions Joint/marginal/conditional, 2D LOTUS, covariance and correlation, Multinomial, Multivariate Normal B&H §7 def →
vizCovariance & Correlation vizThe Multivariate Normal vizMultinomial: Marginals & Lumping vizIndependence of Events vizPairwise vs. Mutual Independence
Transformations Change of variables, convolutions, Beta, Gamma, Beta–Gamma connections, order statistics B&H §8 def →
vizChange of Variables & the Jacobian vizConvolution: Sum of Two R.V.s vizConvolution: Uniform + Exponential vizThe Beta Distribution vizThe Gamma Distribution
Conditional Expectation Given an event, given an r.v., conditional variance, Adam's law and Eve's law, projection view B&H §9 def →
Inequalities & Limit Theorems Cauchy–Schwarz, Jensen, Markov, Chebyshev; LLN; CLT; Chi-Square and Student-t B&H §10 def →
Markov Chains Markov property and transition matrix, classification of states, stationary distribution, reversibility B&H §11
vizMarkov Chains
Markov Chain Monte Carlo Metropolis–Hastings, Gibbs sampling — sampling from distributions you cannot compute B&H §12
Poisson Processes One dimension; conditioning, superposition and thinning; multiple dimensions B&H §13 def →
Brzeźniak & Zastawniak — Basic Stochastic Processes harder follow-on Up next
Topics added as I begin this book.
Statistical Inference
GlossaryProbability Glossary → SourcesCatalog → Practicenone yet
Stanford STATS 200 — Introduction to Statistical Inference Zhou Fan · course notes Up next
Estimators & Sampling Distributions Statistics as random variables; sampling distribution; bias, variance and MSE STATS 200 def →
Maximum Likelihood Estimation Likelihood, log-likelihood, invariance — the same object as minimising cross-entropy STATS 200
Sufficiency & Fisher Information Sufficient statistics, Fisher information, the Cramér–Rao lower bound STATS 200
Efficiency & the MVUE Unbiased estimators, Rao–Blackwell, attaining the CRB — what Kay applies to signals STATS 200
Confidence Intervals Pivotal quantities, coverage, and what a confidence interval does not mean STATS 200 def →
Hypothesis Testing Neyman–Pearson, likelihood-ratio tests, p-values, power — becomes Detection Theory in Kay STATS 200
toolStatistical Tests
Bayesian Inference Priors, posteriors, conjugacy, MAP — and the bridge to regularisation STATS 200 def →
Resampling & Model Selection Bootstrap, jackknife, cross-validation — how deep learning actually measures itself STATS 200
Differential Equations
Arnold — Ordinary Differential Equations · 3Blue1Brown — Differential Equations
Completed
ODE Overview Slope fields, solution curves, initial conditions, order and linearity Arnold · 3B1B viz →
The Pendulum SHM, phase portrait, separatrix, nonlinear period corrections Arnold · 3B1B viz →
Heat Equation ∂u/∂t = α ∂²u/∂x² — Fourier modes, diffusion, decay rates 3B1B viz →
To do — 3Blue1Brown
Fourier Series Decomposing functions into sine/cosine modes; why they solve the heat equation 3B1B
Laplace Transforms Solving ODEs via algebraic manipulation in s-domain 3B1B
To do — Arnold
Phase Portraits & Vector Fields Qualitative analysis of autonomous systems; equilibria and orbits Arnold §1–2
Stability of Equilibria Lyapunov stability; linearisation; stable vs unstable fixed points Arnold §3
Linear ODE Systems x' = Ax; eigenvalue classification of phase portraits Arnold §4
Limit Cycles Isolated closed orbits; Poincaré–Bendixson theorem Arnold §5
Structural Stability & Bifurcations How phase portraits change as parameters vary Arnold §6
Differential Equations on Manifolds Flows on surfaces; torus; global qualitative behaviour Arnold §7–8
Real Analysis
Abbott — Understanding Analysis (Springer UTM, 2nd ed.)
The Real Numbers Axiom of completeness, Archimedean property, uncountability of ℝ Abbott §1
Sequences & Series Convergence, Cauchy sequences, monotone convergence, infinite series Abbott §2
Basic Topology of ℝ Open and closed sets, compactness, connectedness — the real line version before Armstrong's general treatment Abbott §3
Functional Limits & Continuity ε–δ definition, uniform continuity, extreme value and intermediate value theorems Abbott §4
The Derivative Rigorous definition, mean value theorem, L'Hôpital's rule Abbott §5
Sequences & Series of Functions Pointwise vs uniform convergence; power series; Taylor series with rigorous remainder Abbott §6
The Riemann Integral Darboux definition, integrability criteria, fundamental theorem of calculus Abbott §7
Topology
Armstrong — Basic Topology (Springer UTM) · Abbott §3 provides the ℝ warm-up
Topological Spaces & Continuity Open sets, neighbourhoods, continuous maps — without distance Armstrong §1–2
Homeomorphisms Topological equivalence; invariants that tell spaces apart Armstrong §2
Connectedness Path-connected vs connected; intermediate value theorem revisited Armstrong §3
Compactness Open covers; Heine–Borel; why compact spaces behave nicely Armstrong §3
Identification Spaces & Quotient Topology Gluing constructions; torus, Möbius band, Klein bottle Armstrong §4
The Fundamental Group π₁ — loops based at a point; homotopy; simply connected spaces Armstrong §5
Classification of Surfaces Genus, orientability; every compact surface is a sphere, connected sum of tori, or projective planes Armstrong §7
Simplicial Homology Triangulations; boundary operator; H₀, H₁, H₂ — holes in different dimensions Armstrong §8
Applications & Engineering
Dynamics & Orbital Mechanics
GlossaryOrbital Mechanics Glossary → SourcesCatalog → PracticeClassical Mechanics →
Thomson — Introduction to Space Dynamics Dover Current
Kinematics
Circular Motion v = rω, centripetal and tangential acceleration Thomson §1
vizCircular Motion
Angular Velocity Why ω is orthogonal, and v = ω × r Thomson §1
vizAngular Velocity
Plane Motion Radial and transverse components in polar coordinates Thomson §1
vizPlane Motion
Instantaneous Center The point about which the body rotates instantaneously Thomson §1
vizInstantaneous Center
Euler Angles ψ, θ, φ and transformation of displacements Thomson §1
vizEuler Angles
Areal Rate r × v, equal areas, and Kepler's second law Thomson §2
vizAreal Rate def →
Orbital Mechanics
Kepler's First Law Orbits are conic sections — Newton's inverse-square law via Binet's substitution Thomson §4
vizKepler's First Law vizConic Sections def →
Kepler's Second Law Equal areas in equal times — conservation of h = r²θ̇ Thomson §4
vizKepler's Second Law def →
Kepler's Third Law T² = 4π²a³/K — period depends only on semi-major axis Thomson §4
vizKepler's Third Law
Satellite Orbits Radial & transverse equations, Binet's substitution, orbit as a conic Thomson §4
vizSatellite Orbits vizEllipse Geometry def →
Two-Body Problem CoM reduction, relative coordinate, and reduced mass Thomson §4
vizTwo-Body Problem
Orbit from Initial Conditions χ, eccentricity from v₀ and β₀, Fig 4.9-2 chart Thomson §4.9
vizOrbit from Initial Conditions def →
Hohmann Transfer Cotangential transfer between coplanar circular orbits, Δv₁, Δv₂ Thomson §4
vizHohmann Transfer def →
Repulsive Inverse-Square Force F = +K/r² produces repulsive hyperbola — Rutherford scattering Thomson §4.13
vizRepulsive Inverse-Square Force
Impulsive Orbit Change Δv at θ* transfers between orbits sharing the apse line Thomson §4.13
vizImpulsive Orbit Change def →
Orbit from Burnout Given χ, β₀, r₀/R at engine cutoff — derive e, a/R, θ₀ Thomson §4.13
vizOrbit from Burnout def →
Gyrodynamics
Polhode & Herpolhode Motion of the instantaneous rotation axis in body and space frames Thomson §5
vizPolhode & Herpolhode
Rigid Body Dynamics & Gyrodynamics
Euler's Equations for Rigid Body Rotation Body-frame equations of motion; torque-free case Thomson §3
vizMoment of Momentum
Moment of Inertia Tensor Principal axes, eigenvalues of I, Steiner's theorem Thomson §3
vizBar Inertia & Angular Momentum
Precession & Nutation Spinning top under gravity; steady precession conditions Thomson §5
Gyroscope Dynamics Gyroscopic couple, stabilisation, spacecraft attitude control Thomson §5
Electronics & Circuits
Sourcesnone yet PracticeBench track →
The circuit theory behind the Bench track's early rungs (DC fundamentals → analog).
Linear circuits
DC Circuit Analysis Ohm's & Kirchhoff's laws, series/parallel combination, voltage & current dividers, source and load
Network Theorems Thévenin & Norton equivalents, superposition, output impedance, maximum power transfer
Capacitors, Inductors & Transients Reactance, RC/RL first-order response, time constants, charging and step response
AC Steady State & Phasors Sinusoidal steady state, complex impedance, phasor analysis, real and reactive power
Frequency Response & Filters RLC resonance and Q, Bode plots, passive RC/LC filters, corner frequency
Active devices
Diodes & Rectifiers The pn junction, half/full-wave rectifiers and smoothing, Zener references, clamps and limiters
Bipolar Transistors Biasing, common-emitter and emitter-follower amplifiers, the small-signal model
Field-Effect Transistors MOSFETs and JFETs as switches and amplifiers, transfer characteristics, biasing
Operational Amplifiers The ideal op-amp, negative feedback, inverting/non-inverting, active filters, comparators
Oscillators & Signal Generation The Barkhausen criterion, LC and RC feedback oscillators, relaxation (555), crystal oscillators
Signals & Systems
SourcesCatalog → CourseMIT OCW 6.003 →
Oppenheim & Willsky — Signals and Systems (2nd ed.)
Continuous-time signals & systems
Signals & Systems — Classification Memory, causality, stability, invertibility; continuous vs discrete; energy and power signals O&W §1
CT LTI Systems & Convolution Convolution integral, impulse response, BIBO stability, causality from h(t) O&W §2
Fourier Series (CT) Orthogonality, synthesis and analysis equations, Gibbs phenomenon, Parseval's theorem O&W §3
Continuous-Time Fourier Transform CTFT pair, convolution theorem, Parseval, duality; spectrum of standard signals O&W §4
Frequency Response of CT LTI Systems H(jω), ideal filters, first- and second-order system frequency response O&W §3–4
Sampling Theorem Nyquist rate, aliasing, reconstruction; the bridge from CT to DT O&W §7
Laplace & Z-transforms (system perspective)
Laplace Transform & Transfer Functions Region of convergence, poles and zeros, partial fractions, system stability from pole locations O&W §9
Block Diagrams & System Interconnections Series, parallel, feedback; unilateral Laplace for IVPs; signal flow graphs O&W §9–10
Z-Transform ROC, poles and zeros, inverse Z-transform; DT system stability O&W §10
Signal Processing
Oppenheim & Schafer — Discrete-Time Signal Processing
Sampling & Reconstruction Nyquist theorem, aliasing, reconstruction from samples Oppenheim §1
Convolution & LTI Systems Impulse response, linearity, time-invariance, BIBO stability Oppenheim §2
Z-Transform Region of convergence, poles and zeros, inverse Z-transform Oppenheim §3
Discrete Fourier Transform & FFT DFT as sampled spectrum; FFT algorithm; spectral leakage and windowing Oppenheim §8
Estimation & Filtering
Kay — Statistical Signal Processing (Estimation & Detection) · builds on Probability
Estimation & detection
Estimation Theory MVUE, Cramér–Rao bound, maximum likelihood estimation, bias vs variance Kay Vol.1 §1–3
Detection Theory Hypothesis testing, Neyman–Pearson, ROC curves, matched filter Kay Vol.2 §1–3
Wiener Filter Optimal linear filter for stationary processes; MMSE estimation Kay Vol.1 §12
Kalman & sequential filtering
Linear Kalman Filter State-space model, predict–update cycle, optimal gain, covariance propagation Kay · Welch & Bishop tutorial
Extended & Unscented Kalman Filter Linearisation for nonlinear systems; UKF sigma-point approach; orbit determination Kay · Crassidis & Junkins
Particle Filters Sequential Monte Carlo; non-Gaussian state estimation; tracking highly nonlinear dynamics Arulampalam et al. tutorial
Control Systems
Ogata — Modern Control Engineering (5th ed.) · Franklin — Feedback Control of Dynamic Systems
Classical control
Mathematical Models of Systems Transfer functions from ODEs; block diagram algebra; linearisation around operating points Ogata §2
Time-Domain Response First- and second-order step response; damping ratio, natural frequency, settling time, overshoot Ogata §4
Root Locus Closed-loop poles as gain varies; construction rules; designing for desired transient specs Ogata §6
Frequency Response & Bode Plots Magnitude and phase vs frequency; asymptotic Bode construction; gain and phase margins Ogata §7
Nyquist Stability Criterion Encirclement condition; stability margins in the frequency domain; robustness Ogata §8
PID Controllers Proportional, integral, derivative action; Ziegler–Nichols tuning; practical limitations Ogata §8
Modern (state-space) control
State-Space Representation ẋ = Ax + Bu, y = Cx + Du; eigenvalues as poles; solution via matrix exponential Ogata §9
Controllability & Observability Controllability matrix rank; observability matrix rank; duality theorem Ogata §10
State Feedback & Pole Placement Full-state feedback u = −Kx; placing closed-loop poles; integral action for zero steady-state error Ogata §10
State Observers (Luenberger) Estimating states from outputs; observer gain; separation principle — direct precursor to Kalman filter Ogata §10 · Franklin §7
LQR & Optimal Control Quadratic cost minimisation; Riccati equation; relationship between LQR and Kalman filtering (LQG) Franklin §9
RF & Communications
Proakis & Salehi — Communication Systems Engineering (2nd ed.) · Haykin — Communication Systems
Foundations
Channel Models & Noise AWGN channel, SNR, noise power spectral density; thermal noise and N₀/2 Proakis §3
Analog Modulation AM, DSB-SC, SSB, FM, PM; bandwidth and power tradeoffs; coherent vs envelope detection Proakis §3
Digital communications
Digital Modulation — BPSK, QPSK, QAM Signal constellations, decision regions, BER vs Eb/N₀; matched filter receiver Proakis §5
Spread Spectrum — DSSS & FHSS PN sequences, processing gain, jamming margin; CDMA; GPS L1 C/A signal structure Proakis §9
Channel Coding & Shannon Capacity Shannon's theorem, capacity–bandwidth tradeoff; Hamming codes; convolutional codes and Viterbi Proakis §8
RF propagation & link budgets
Antenna Fundamentals Gain, directivity, effective aperture, beam pattern; dipole and patch antennas for GNSS receivers Proakis §2 · Balanis §1–2
Link Budget & Friis Equation EIRP, path loss, receiver sensitivity, C/N₀; computing GPS received power from 20,200 km Proakis §2
Multipath & Ionospheric Effects Rayleigh fading, Doppler shift, ionospheric delay model; how these degrade GNSS pseudorange Proakis §13 · Kaplan §7
GNSS & Navigation
Kaplan & Hegarty — Understanding GPS/GNSS · Vallado — Fundamentals of Astrodynamics
GNSS Architecture & Signal Structure GPS/Galileo/GLONASS signal design; pseudorange, carrier phase, satellite geometry (DOP) Kaplan & Hegarty §1–4
Spoofing, Jamming & Resilient PNT Threat taxonomy; detection algorithms; alternative and complementary positioning (IMU, eLoran, LEO PNT) Kaplan §9 · CISA advisories
Orbit Determination & SSA Batch least squares, sequential estimation, conjunction analysis — Thomson dynamics + Kalman estimation combined Vallado · Crassidis & Junkins
Deep Learning
GlossaryDeep Learning Glossary → SourcesCatalog → Practicenone yet
Prince — Understanding Deep Learning MIT Press 2024 · free at udlbook.github.io Current
Foundations
Supervised Learning Input–output mappings, loss functions, empirical risk minimisation Prince §2
vizPerceptron
Shallow Neural Networks One hidden layer, activation functions, universal approximation Prince §3 def →
vizShallow Neural Network vizWhy ReLU?
Activation Functions ReLU, sigmoid, tanh — the nonlinearity that makes depth worth having Prince §3 def →
Deep Neural Networks Composing layers, depth vs width, piecewise linear regions Prince §4 def →
Loss Functions MSE, cross-entropy, maximum likelihood perspective Prince §5 def →
vizMulti-class Cross-Entropy
Training
Gradient Descent & Backpropagation SGD, mini-batch, chain rule through computation graphs Prince §6 def →
vizBackpropagation
Gradients & Initialisation Vanishing/exploding gradients, He/Xavier init, batch norm Prince §7 def →
Measuring Performance Bias–variance tradeoff, train/val/test split, double descent Prince §8
vizNoise, Bias & Variance vizA Simplified Network (Fig 8.4) vizDouble Descent vizInductive Bias vizThe Curse of Dimensionality
Regularisation L1/L2, dropout, data augmentation, early stopping Prince §9
Architectures
Convolutional Networks Convolution, pooling, receptive field, translation equivariance Prince §10 def →
vizConvolutional Networks vizChannels in CNNs vizWhy Multiple Output Channels?
Residual Networks Skip connections, batch norm, modern CNN architectures Prince §11
Transformers Self-attention, multi-head attention, positional encoding, encoder–decoder Prince §12
vizDot-Product Self-Attention
Graph Neural Networks Message passing, node/edge/graph classification, relational inductive bias Prince §13
Generative models
Unsupervised Learning Clustering, dimensionality reduction, self-supervised pretraining Prince §14
Generative Adversarial Networks Minimax game, training instability, mode collapse, Wasserstein GAN Prince §15
Normalizing Flows Bijective mappings, change-of-variables formula, exact likelihood Prince §16
Variational Autoencoders Latent variable models, ELBO, reparameterisation trick Prince §17
Diffusion Models Forward noising process, denoising score matching, DDPM Prince §18
Reinforcement learning
Reinforcement Learning MDP, policy/value functions, Q-learning, policy gradient, RLHF Prince §19
Practicals hands-on projects to accompany the theory
Backprop from scratch 2-layer MLP on toy 2D data; verify gradients numerically
Train a small CNN on MNIST From scratch in PyTorch; reach >98% test accuracy
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