The previous note stated the Gram-Schmidt algorithm and worked an example, but never proved it always succeeds. This sub-note carries the inductive proof: at every step, the newly constructed vector really is orthogonal to every previous one, and the running list really does span the same subspace as the corresponding prefix of the original basis.
Learning objectives. By the end of this note you should be able to:
- state the two-part inductive claim Gram-Schmidt's correctness rests on;
- prove the orthogonality claim at each step;
- prove the span-preservation claim at each step;
- conclude the full theorem by induction.
In this lesson
- The inductive claim
Before you start
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