Linear Algebra & Optimization
Linear algebra and optimization underpin portfolio construction, factor models and machine learning.
Lessons · recommended order
- 1Vectors & Dot ProductsVectors encode direction and magnitude; the dot product measures alignment and drives projections, correlation and factor exposures.Foundations7 min
- 2Matrices & Linear MapsA matrix is a linear map between vector spaces; matrix multiplication composes maps, and inverses undo them when they exist.Foundations8 min
- 3Eigenvalues & EigenvectorsEigenvectors are directions a matrix only stretches, and eigenvalues are the stretch factors — the key to understanding repeated application of a linear map.Intermediate8 min
- 4Rank-Nullity TheoremThe rank-nullity theorem splits the domain of a linear map into what survives (column space) and what collapses to zero (null space).Intermediate6 min
- 5Orthogonality & ProjectionsProjecting a vector onto a subspace finds the closest point in that subspace — the geometric heart of least squares and hedging.Intermediate7 min
- 6Positive Definite MatricesA matrix is positive definite when every quadratic form it induces is positive — the algebraic signature of a valid variance or a strict minimum.Intermediate7 min
- 7Covariance MatricesThe covariance matrix packages every pairwise variance and covariance among assets into a single PSD matrix that drives portfolio risk.Intermediate7 min
- 8Principal Component AnalysisPCA finds the directions of maximum variance in data by solving a constrained optimization whose answer is the top eigenvector of the covariance matrix.Advanced8 min
- 9Lagrange MultipliersLagrange multipliers turn constrained optimization into an unconstrained stationarity condition by pricing the constraint with a multiplier.Advanced7 min
- 10Convexity & the HessianA function is convex when its Hessian is PSD everywhere, guaranteeing that any local minimum is a global one — the key to trusting an optimizer's answer.Advanced7 min