A subset 
of a vector space 
, with the inner product 
, is called orthonormal if 
when 
. That is, the vectors are mutually perpendicular.
Moreover, they are all required to have length one: 
.
An orthonormal set must be linearly independent, and so it is a vector basis for the space it spans. Such a basis is called an orthonormal basis.
The simplest example of an orthonormal basis is the standard basis 
for Euclidean space 
. The vector 
is the vector with all 0s except for a 1 in the 
th coordinate. For example, 
. A rotation (or flip) through the origin will
send an orthonormal set to another orthonormal set. In fact, given any orthonormal
basis, there is a rotation, or rotation combined with a flip, which will send the
orthonormal basis to the standard basis. These are precisely the transformations
which preserve the inner product, and are called orthogonal
transformations.
Usually when one needs a basis to do calculations, it is convenient to use an orthonormal basis. For example, the formula for a vector space projection is much simpler with an orthonormal basis. The savings in effort make it worthwhile to find an orthonormal basis before doing such a calculation. Gram-Schmidt orthonormalization is a popular way to find an orthonormal basis.
Another instance when orthonormal bases arise is as a set of eigenvectors for a symmetric matrix. For a general matrix, the set of eigenvectors may not be orthonormal, or even be a basis.
