Orthonormal Sets and Bases

An orthogonal set is called an orthonormal set if for .

If an orthonormal set is a basis for the subspace of , then is an orthonormal basis for .

Orthonormal sets are just orthogonal sets with magnitude 1

The standard basis for is an orthonormal set (and an orthonormal basis for ). The set

is also an orthonormal set (and also an orthonormal basis for ).

Given an orthogonal basis for a subspace of , we can obtain an orthonormal basis

for by letting

for .

That is, we divide each vector by its magnitude.