A book of holdings is a vector; a list of cash-flow or factor exposures is a matrix; matching a target payoff or a balance-sheet constraint is solving . This note rebuilds the linear-algebra core at the level needed for those desk jobs — vectors and the dot product, matrix algebra and the column picture of , exact linear systems, inverses, and rank / null space — without replaying the whole of Linear Algebra I. The result is a single language for holdings, a correctly centred return matrix and a small replicating system. Each row and column gets a financial meaning before we solve it.
Learning objectives. By the end of this note you should be able to:
- compute dot products, matrix–vector products, and basic matrix algebra with finance labels attached;
- set up and solve a small (unique, none, or infinitely many);
- read invertibility, rank, and the null space as statements about holdings and replication;
- recognise a sample Gram matrix as the prototype of a covariance-ready object.
In this lesson
- Holdings as vectors
- Matrices and the product
- Linear systems
The full lesson, with worked examples and exercise solutions, is for members.
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