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Linear Algebra, Optimisation and Numerical Methods / Vector Spaces, Subspaces, and the Four Fundamental Subspaces

Derivation — Proof of the Steinitz Exchange Lemma

Theorem 1 of the previous note stated the Steinitz exchange lemma without proof, because the proof — while short — is exactly the kind of algebra that would break the reading pace of a note trying to establish why dimension matters. This sub-note supplies it: an explicit procedure that trades spanning vectors for independent ones, one at a time, and shows the trade can never run out of vectors to swap before the independent set is exhausted. That the procedure can always complete is the entire content of the lemma.
Learning objectives. By the end of this note you should be able to:
  • carry out one step of the exchange argument explicitly;
  • explain why each step preserves the spanning property;
  • follow the induction that completes the proof for general kk and mm.

In this lesson

  1. The exchange step
  2. Iterating the exchange
  3. Corollary — the dimension is well defined

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