7 Orthogonality

In defining a vector space we kept only addition and scaling, and deliberately discarded length and angle. This section puts them back.

An inner product is the extra structure required. From it come the norm, the Cauchy–Schwarz inequality, and a meaning for two vectors being perpendicular. Once perpendicularity is available, two constructions follow: the Gram–Schmidt procedure, which converts any basis into an orthonormal one, and the decomposition \(V=A\oplus A^{\perp }\), which splits every vector into a part lying in a chosen subspace and a part perpendicular to it.

That decomposition is the section’s real content, and it is more useful than it looks. The component lying in the subspace is the point of the subspace closest to the original vector, which is why the same picture underlies least squares approximation, and why orthonormal bases are worth the effort of constructing.

Questions on this section

Stuck on something here? Ask below and it stays attached to this topic.