where is the Kronecker delta function and is the inner product (which can also be extended to real functions; as such a set of functions can be orthonormal). The span of the set forms an orthonormal basis. An example of this is Cartesian coordinates.
A matrix whose columns form an orthonormal set is called an orthogonal matrix, and will have the property
where is the inverse of and is its transpose. Real orthogonal matrices will have a determinant of either 1 or -1. An extension to complex matrices is a matrix whose inverse is equal to its conjugate transpose, and is called a unitary matrix.