Part VII: Applied mathematics and computation
Linear algebra, probability, statistics, numerical methods, and differential equations, put to work in systems, GPUs, machine learning, robotics, and aerospace.
- A01Linear systems, vector spaces, and transformations5 exercisesInterpret a matrix as a map, solve and classify linear systems, identify span, independence, basis, rank, and null space, and explain when an inverse or determinant is useful.
- A02Orthogonality, least squares, eigenvectors, and SVD5 exercisesProject onto a subspace, derive least squares from residual orthogonality, compute a small eigendecomposition, recognize positive definiteness, and interpret singular values geometrically.
- W04Geometry, gradients, and automatic differentiation5 exercisesDerive gradients with correct shapes; implement forward-mode differentiation from an algebraic rule; distinguish exact symbolic structure, floating-point evaluation, finite differences, and automatic differentiation.
- A03Probability, random variables, and simulation5 exercisesCompute conditional probabilities without reversing them, distinguish independence from exclusivity, derive expectations and variances, and choose a probability model from assumptions rather than appearance.
- A04Statistics, experiments, and uncertainty in measurements5 exercisesDistinguish a population parameter from an estimate, interpret confidence intervals and tests correctly, design a paired experiment, and report effect sizes and limitations instead of a single persuasive number.
- A05Numerical error, conditioning, roots, and approximation5 exercisesSeparate problem sensitivity from algorithmic error, bracket a root safely, compare differentiation methods, and verify a numerical answer with residuals, scale, and known solutions.
- A06Differential equations, stability, and optimization5 exercisesConvert a differential equation into state form, carry out Euler and RK4 steps, test convergence against a known solution, analyze a simple stability restriction, and distinguish gradient descent from Newton optimization.
- W05From differential equations to trustworthy simulation5 exercisesDerive two discrete methods from a continuous model; analyze stability and energy separately; validate against an analytic trajectory and a step-halving experiment.
- T01Systems and performance engineering5 exercisesExplain a data-layout hypothesis, measure a bounded kernel, separate algorithmic work from machine effects, and choose a reliability/performance tradeoff with evidence.
- T02GPU, parallel, and graphics computing5 exercisesDecompose a reduction, reason about work/span and data transfer, explain basic GPU execution and memory scopes, and validate a CPU alternative before attempting optional CUDA execution.
- W06Linear algebra becomes graphics and spatial computation5 exercisesKeep coordinate spaces and homogeneous conventions explicit; derive barycentric and perspective-correct interpolation; connect geometric correctness to numerical and performance evidence.
- T03Machine learning, gradients, evaluation, and attention5 exercisesDerive logistic-regression gradients, check a tiny network’s backpropagation, keep evaluation data separate, and explain the shapes and limited meaning of a small attention calculation.
- T04Embedded software, robotics, feedback, and state estimation5 exercisesAnalyze a sampled feedback loop, implement a simulated PID controller with saturation, derive a scalar estimation update, and distinguish simulation evidence from real-time or safety validation.
- T05Scientific and aerospace-related computing5 exercisesConstruct a nondimensional motion model, derive conserved quantities, validate a numerical orbit against an analytic solution, and distinguish model error from discretization and roundoff error.
- chapter
- workshop: connects two subjects
- checkpoint, review or assessment