# Orthonormal Vectors

##### 143 words — categories: linear-algebra

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.