# Reflection Operators

Definition: For any vector $\vec{x} \in \mathbb{R}^2$, a reflection transformation operator $T: \mathbb{R}^n \to \mathbb{R}^n$ reflects every vector $\vec{x}$ to its symmetric image about some line. For any vector $\vec{x} \in \mathbb{R}^3$, a reflection transformation operator reflects every vector $\vec{x}$ to its symmetric image about some plane ($\mathbb{R}^3$). |

Let's first look at some reflection operators in $\mathbb{R}^2$ and then subsequently in $\mathbb{R}^3$.

## Reflection Transformations in 2-Space

Let $\vec{x} \in \mathbb{R}^2$ such that $\vec{x} = (x, y)$ and suppose that we want to reflect $\vec{x}$ across the $y$-axis as illustrated:

Thus the $x$-coordinate of our vector will be the opposite to that of our image. The following equations summarize our image:

(1)Thus our standard matrix is $A = \begin{bmatrix} -1 & 0\\ 0 & 1 \end{bmatrix}$, and in $w = Ax$ form we get that:

(2)Of course there are other types of reflection transformations in $\mathbb{R}^2$ such as reflecting across the $x$-axis, as well as the diagonal line $y = x$. The table below illustrates these transformations alongside their associated standard matrices. It is good to verify where these standard matrices arise.

Operator | Visual | Equations Defining the Image | Standard Matrix |
---|---|---|---|

Reflection about the $x$-axis | $w_1 = x + 0y \\w_2 = 0x -y$ | $\begin{bmatrix} 1 & 0\\ 0 & -1 \end{bmatrix}$ | |

Reflection about the line $y = x$ | $w_1 = 0x + y \\w_2 = x + 0y$ | $\begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}$ |

## Reflection Transformations in 3-Space

Like in $\mathbb{R}^2$, we can take some vector $\vec{x} = (x, y, z)$ and reflect it. This time we will be reflecting over planes instead of lines however. For example, consider a vector $\vec{x} \in \mathbb{R}^3$ that is reflected about the $xy$-plane as illustrated in the following diagram:

.

We note that the only difference between $\vec{x}$ and its image is the sign change in the $z$-coordinate, and thus, the following equations define our image vector:

(3)Thus our standard matrix is $A = \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & -1 \end{bmatrix}$, and in $w = Ax$ form we get:

(4)The following table describes some reflections across some other common planes:

Operator | Equations Defining the Image | Standard Matrix |
---|---|---|

Reflection across the $xz$-plane | $w_1 = x + 0y + 0z \\ w_2 = 0x -y + 0z \\ w_3 = 0x + 0y + z$ | $\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{bmatrix}$ |

Reflection across the $yz$-plane | $w_1 = -x + 0y + 0z \\ w_2 = 0x +y + 0z \\ w_3 = 0x + 0y + z$ | $\begin{bmatrix}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$ |