---
tag: 0VMWGYY35Q
kind: statement
role: theorem
name: Cartan–Dieudonné theorem
msc: [51F15, 15B10]
uses: [025158VEYZ, 0QRBVFJ50X]
sources:
  - url: https://en.wikipedia.org/wiki/Cartan–Dieudonné_theorem
    locator: Formal statement
same_as:
  wikidata: Q2540738
  wikipedia: Cartan–Dieudonné theorem
---

## Statement

Let $V$ be a real inner product space of finite dimension $n$.
Then each linear isometry $T \colon V \to V$ is a composition of at most $n$ [[reflections|06BRDF3F0E]] in hyperplanes of $V$.

## Remarks

The identity map is the composition of zero reflections.
In $\R^2$, each rotation is the composition of two reflections in lines through the origin.
The theorem holds more generally for nondegenerate symmetric bilinear forms on spaces over fields of characteristic not $2$.

## Explanations

### L1 human

Every rigid linear motion of $n$-dimensional space is a sequence of at most $n$ mirror reflections.
So reflections are the building blocks of all isometries.
In the plane, a rotation by $\theta$ is the product of two reflections in lines at the angle $\theta/2$.