Example 2 of the span note and Exercise 4 of the column-picture note both flagged, informally, that an instrument can be "redundant" — replicable from the others, adding no new risk. This note makes that idea precise with linear independence, builds the notion of a basis — a minimal, redundancy-free spanning set — and states the Steinitz exchange lemma, the single fact that makes "the dimension of a subspace" a well-defined number rather than an artefact of which spanning set you happened to write down. Every rank, every dimension count quoted anywhere else in this unit rests on this lemma.
Learning objectives. By the end of this note you should be able to:
- test a set of vectors for linear independence, both by definition and via the pivot count of their matrix;
- define a basis and verify a proposed basis is both spanning and independent;
- state the Steinitz exchange lemma and explain why it guarantees dimension is well defined;
- compute the dimension of a subspace given in span or nullspace form.
In this lesson
- Linear independence
- Basis
- The Steinitz exchange lemma
Before you start
The full lesson, with worked examples and exercise solutions, is for members.
Read the full lessonNew to SelfMadeQuant.com? See membership options or browse the Linear Algebra, Optimisation and Numerical Methods syllabus.