Why linear algebra is the language of finance

Almost every object a working quant manipulates is a vector, a matrix, or a function of one. A portfolio of n assets is a weight vector w in Rⁿ. Its risk is a quadratic form wᵀΣw in the covariance matrix Σ. A factor model decomposes returns as Bf + ε, a linear map plus noise. A regression is a projection. Principal components are eigenvectors of a covariance matrix. The Kalman filter is two matrix multiplications and one inverse.

This course teaches the linear algebra you actually need to read modern factor-model papers, build a Markowitz optimisation, decompose a yield curve, simulate correlated returns, or debug why your covariance matrix isn't invertible at 4 a.m. on a trading floor. It is rigorous where rigour pays — the spectral theorem, the SVD, the projection theorem — and pragmatic everywhere else.

Four problems linear algebra solves for a quant

1. Pricing a portfolio's risk: the variance of wᵀr is wᵀΣw — a single quadratic form that hides an entire industry of statistical estimation.
2. Decomposing returns into a few common factors: SVD or PCA, the mathematical core of every quant equity book at Two Sigma or Citadel.
3. Solving for the minimum-variance or maximum-Sharpe portfolio: a constrained quadratic program that, in its simplest form, has a closed-form linear-algebraic solution.
4. Simulating correlated returns: a Cholesky factorisation of Σ turns independent normal noise into a path that respects observed dependencies.

How this course is built

• Modules 2-4 build the vocabulary: vectors, matrices, linear systems.
• Modules 5-6 cover subspaces, rank, projections, least squares — the geometric heart of regression.
• Modules 7-9 develop the eigen/spectral/SVD machinery — the engine of factor models, PCA, and risk decomposition.
• Modules 10-11 apply that machinery to yield curves and matrix calculus.
• Module 12 closes with the numerical-analysis caveats that turn pristine algebra into reliable code.