Overview — Linear Systems and Least Squares

Every portfolio weight vector, every hedge ratio, and every factor loading is the solution of a linear system — or of a least-squares problem when the system is overdetermined. This topic rebuilds the linear-algebra toolkit at the level a quant actually uses: holdings as vectors, covariance-ready matrices, exact solves of $A\mathbf{x}=\mathbf{b}$, orthogonal projection for regression, and the eigenbasis of a covariance matrix as the variance basis. It synthesises material that used to live across Linear Algebra I and II into a single finance-facing arc; the pillar notes remain available for full proofs.

By the end of this note you should be able to:

• solve a small replicating or holdings system and read off rank / null-space consequences;
• set up OLS as projection onto a column space and write the normal equations;
• diagonalise a symmetric covariance and interpret eigenvalues as directional variances.

Read the lessons in order: exact payoff matching comes first, approximate fitting comes next, and the covariance eigenbasis follows. Keep the distinction between an exact algebraic fit and a statistical forecast throughout.

1. Contents
2. The big picture