orthogonality

Summary

Orthogonal set

• vectors are pairwise orthogonal

can include the 0 vector

Orthonornal set

• orthogonal and normalised

magnitude of 1

Algorithm to check for orthogonality to a subspace

Concept

Orthogonal vectors

ie. either is the zero vector or two vectors are perpendicular

Orthogonal to subspaces

• perpendicular to every vector

• relation to nullspace of the matrix transpose

is a matrix of the basis of , simultaneously

Orthogonal complement

• set of all orthogonal vectors

Application

Normalising an orthogonal set

Vector orthogonal to a subspace

hence the transpose before the nullspace, due to the inner product

Orthogonality of hyperplanes

• span of the normal makes up the orthogonal complement of a plane

Rowspace is orthogonal to nullspace