---
tag: 025158VEYZ
kind: definition
name: Reflection
msc: [15A04, 51F15]
defines: 06BRDF3F0E
uses: [0Q8HZTJC7G]
sources:
  - url: https://en.wikipedia.org/wiki/Reflection_(mathematics)
    locator: Reflection through a hyperplane in n dimensions
---

## Definition

Let $V$ be an inner product space over $\R$ or $\C$, and let $W$ be a finite-dimensional subspace of $V$, with [[orthogonal projection|0B8KZRT28M]] $P_W$.
The *reflection* in $W$ is the linear map
$$R_W \colon V \to V, \qquad R_W(v) = 2 P_W(v) - v.$$

## Remarks

$R_W$ fixes each vector of $W$, negates each vector of $W^\perp$, and satisfies $R_W^2 = \id_V$.
It is a linear isometry.
For a hyperplane $W = a^\perp$ with $a \ne 0$, the formula is $R_W(v) = v - 2 \frac{\langle v, a \rangle}{\langle a, a \rangle} a$.
In $\R^n$, the matrix of a hyperplane reflection is orthogonal with determinant $-1$.

## Explanations

### L1 human

A reflection is a mirror image in a subspace.
Vectors in the mirror stay where they are, and vectors perpendicular to it change direction.
Reflecting twice brings every vector back.