A collection of vectors \(a_1, \cdots, a_k\) is orthogonal, or mutually orthogonal if \(a \perp b\) for any \(i\), \(j\) with \(i \neq j\), \(i\), \(j = 1, \cdots, k\). A collection of vectors \(a_1, \cdots, a_k\) is orthonormal if its orthogonal and \(||a_i|| = 1\) for \(i = 1, \cdots, k\).

A vector of norm one is called normalized; dividing a vector by its norm is called normalizing it. Each vector in an orthonormal collection of vectors is normalized, and two different vectors from the collection are orthogonal. These two conditions can be combined into a statement about the inner product of pairs of vectors in the collection: \(a_1, \cdots, a_k\) is orthonormal means that

\[a_i^T a_j = \begin{cases} 1 & i = j \\ 0 & i \neq j \end{cases}\]

Orthonormal vectors are linearly independent, and therefore also an (orthonormal) basis.