Gram-Schmidt Example

Linear Algebra (Matrices) · Axiom Academy

EXAMPLE Gram-Schmidt: building an orthonormal basis Turn three independent vectors in into a perpendicular, unit-length basis, one vector at a time Apply the Gram-Schmidt process to the basis , , of . First build an orthogonal basis by subtracting projections, then normalize it to an orthonormal basis . The goal: three arrows meeting at right angles (the square marks). Gram-Schmidt builds them in order, peeling off each vector's overlap with the ones before it. Nice work. You ran the full Gram-Schmidt process on and produced an orthonormal basis of . One vector at a time: keep , then each new subtracts off its overlap with every earlier , so it lands perpendicular to all of them. The projection formula: — using in the denominator lets you work with the un-normalized . Orthogonalize, then normalize: the result was , ; dividing each by its length gave , , . Span is preserved: every (and ) is a combination of the original 's, so the new basis spans the same space — here, all of . This is the engine behind QR decomposition and least-squares: an orthonormal basis makes projections, distances and coordinates easy to compute.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.