User:Gmh5016/Sandbox

In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal and both of unit length. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. A basis which forms an orthonormal set is called an orthonormal basis.

For example, the standard basis for Euclidean 3-space {i,j,k} is orthonormal, because i&middot;j = 0, j&middot;k = 0, k&middot;i = 0 and each of them is a unit vector.

A set of vectors can be transformed into an orthonormal set by applying the Gram–Schmidt process, and then normalizing each vector.

When referring to real-valued functions, usually the L&sup2; inner product is assumed unless otherwise stated, so two functions $$\phi(x)$$ and $$\psi(x)$$ are orthonormal over the interval $$[a,b]$$ if
 * $$(1)\quad\langle\phi(x),\psi(x)\rangle = \int_a^b\phi(x)\psi(x)dx = 0,\quad{\rm and}$$
 * $$(2)\quad||\phi(x)||_2 = ||\psi(x)||_2 = \left[\int_a^b|\phi(x)|^2dx\right]^\frac{1}{2} = \left[\int_a^b|\psi(x)|^2dx\right]^\frac{1}{2} = 1.$$

An equivalent formulation of the two conditions is done by using the Kronecker delta. A set of vectors (functions, matrices, sequences etc)
 * $$ \left\{ u_1, u_2 , ... , u_n , ... \right\} $$

forms an orthonormal set if and only if
 * $$ \forall n,m \ : \quad \left\langle u_n | u_m \right\rangle = \delta_{n,m} $$

where < | > is the inner product defined over the vector space.

;Category:Linear algebra Category:Functional analysis

Ortonormalita Ortonormal Orthonormalsystem Ortonormal Base orthonormale Ortonormalność Ortonormalidade