---
tag: 0JWM6GJNQW
kind: statement
role: proposition
name: An orthonormal family is linearly independent
msc: [15A63, 46C05]
uses: [09VYHEF2VT, 0EVHJFTG9Z]
sources:
  - url: https://en.wikipedia.org/wiki/Orthonormality
    locator: Properties
---

## Statement

Let $V$ be an inner product space over $\R$ or $\C$.
Then each [[orthonormal family|0RW9TNNT87]] in $V$ is [[linearly independent|0CRMGPAH83]].

## Remarks

If $\sum_i a_i e_i = 0$, then the inner product with $e_j$ gives $a_j = 0$.
More generally, a family of nonzero pairwise orthogonal vectors is linearly independent.
So an orthonormal family in an $n$-dimensional space has at most $n$ vectors.

## Explanations

### L1 human

Vectors at right angles cannot be combinations of each other.
To find the coefficient of $e_j$ in a combination, take the inner product with $e_j$.
If the combination is zero, then every coefficient is zero.