The previous note defined orthogonality abstractly; this note shows why a basis of mutually orthogonal, unit-length vectors is worth building at all — an orthonormal basis turns the general coordinate problem (solve a linear system for the coefficients, Proposition 1 of the LA-I independence note) into a single inner-product computation per coefficient, with no system-solving at all.
Learning objectives. By the end of this note you should be able to:
- verify a set of vectors is orthonormal;
- prove that an orthonormal set is automatically linearly independent;
- read a vector's coordinates off an orthonormal basis directly, with no elimination;
- compute a norm directly from orthonormal coordinates via a generalised Pythagorean identity.
In this lesson
- Orthonormal sets
- Coordinates relative to an orthonormal basis
Before you start
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