This closing note of Topic 1 answers a question every earlier result in the topic has been building toward: given a subspace and a vector not in it, what is the closest point of to ? The answer — the orthogonal projection — is constructed directly from an orthonormal basis of (Corollary 1 of the previous note guarantees one always exists), and the closest-point property is not a side benefit — it is the reason orthogonal projection is the universal mechanism behind least-squares regression, developed fully in the next topic.
Learning objectives. By the end of this note you should be able to:
- construct the orthogonal projection onto a subspace given an orthonormal basis for it;
- prove that the projection is the unique closest point in the subspace;
- decompose any vector into its projection plus an orthogonal remainder;
- state the projection formula as a matrix and recognise it as idempotent.
In this lesson
- Orthogonal complement and projection
- The minimisation property
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