---
tag: 0FWJ0VW561
kind: definition
name: Orthonormal basis
msc: [15A63, 46C05]
defines: 0S9GGTR0EX
uses: [09VYHEF2VT, 0VTQTPQGAV]
sources:
  - url: https://en.wikipedia.org/wiki/Orthonormal_basis
    locator: lead section
---

## Definition

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$.
An *orthonormal basis* of $V$ is a [[basis|0KCQ9Y3QKA]] $(e_1, \dots, e_n)$ of $V$ that is an [[orthonormal family|0RW9TNNT87]].

## Remarks

An orthonormal family of $\dim V$ vectors is an orthonormal basis.
In an orthonormal basis, coordinates are inner products: $v = \sum_i \langle v, e_i \rangle e_i$.
The vectors $\frac{1}{\sqrt 2}(1, 1)$ and $\frac{1}{\sqrt 2}(1, -1)$ form an orthonormal basis of $\R^2$.
In infinite-dimensional Hilbert spaces, the term means a maximal orthonormal family, which is in general not a basis in the algebraic sense.

## Explanations

### L1 human

An orthonormal basis is a set of perpendicular unit vectors that spans the space.
Each coordinate of a vector is its inner product with one basis vector.
No linear system needs to be solved to find coordinates.