A combination of instruments that produces exactly zero payoff in every scenario is not "nothing" — it is a redundancy: proof that some instruments in the set are not adding new risk, and (if it can be bought at nonzero cost) a candidate arbitrage. This note studies exactly that set, the nullspace , and gives the systematic algorithm — reduced row echelon form and free variables — for describing it completely, generalising the redundancy check of Exercise 4 in the column-picture note from a single spotted relationship to an exhaustive, mechanical procedure.
Learning objectives. By the end of this note you should be able to:
- confirm the nullspace is a subspace using the subspace test;
- reduce a matrix to reduced row echelon form (RREF) and identify pivot and free columns;
- construct the special solutions that span the nullspace, one per free variable;
- read the dimension of the nullspace directly off the RREF.
In this lesson
- The nullspace as a subspace
- Reduced row echelon form and free variables
Before you start
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