The previous note showed why an orthonormal basis is worth having — coordinates become a single inner product, variance decomposes with no cross terms. This note supplies the construction: an algorithm, the Gram-Schmidt process, that converts any basis of an inner-product space into an orthonormal one spanning the same space. The method is a repeated application of the exact orthogonal-decomposition idea used inside the Cauchy-Schwarz proof — subtract off the part along already-processed directions, keep only what's left.
Learning objectives. By the end of this note you should be able to:
- run the Gram-Schmidt algorithm by hand on a small basis;
- identify the projection-and-subtract step as the algorithm's core move;
- explain why each output vector lands in the span of the inputs processed so far;
- recognise the -factorisation reading of the algorithm (previewed here, developed fully in the next topic).
In this lesson
- The algorithm
- The preview
Before you start
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