An orthonormal basis of a finite-dimensional vector space 
with inner product 
is a vector basis 
whose vectors
are mutually perpendicular and have length
one. Thus 
for 
and 
.
An orthonormal set is linearly independent, so it is an orthonormal basis for the vector space it spans.
The simplest example of an orthonormal basis is the standard basis 
for Euclidean space 
. The vector 
has all coordinates equal
to 0 except for a 1 in the 
th coordinate. For example,
.
In Euclidean space, a rotation
about the origin, or such a rotation
combined with a reflection through a hyperplane
containing the origin, sends an orthonormal
set to another orthonormal set. Conversely,
any orthonormal basis of 
can be sent to the standard
basis by a rotation, possibly combined with a reflection. These linear
transformations preserve the inner product and
are called orthogonal transformations.
It is often convenient to use an orthonormal basis for calculations. For example, the formula for a vector space projection is simpler in an orthonormal basis. Gram-Schmidt orthonormalization produces an orthonormal basis from a vector basis.
A real symmetric matrix has an orthonormal basis of eigenvectors. For a general matrix, the eigenvectors need not be orthonormal or even form a vector basis.
