# Linear Transformation that Maps Each Vector to Its Reflection with Respect to $x$-Axis

## Problem 597

Let $F:\R^2\to \R^2$ be the function that maps each vector in $\R^2$ to its reflection with respect to $x$-axis.

Determine the formula for the function $F$ and prove that $F$ is a linear transformation.

## Solution 1.

Let $\begin{bmatrix} x \\ y \end{bmatrix}$ be an arbitrary vector in $\R^2$.
Its reflection with respect to $x$-axis is the vector $\begin{bmatrix} x \\ -y \end{bmatrix}$.
Thus the formula for the function $F$ is given by
$F\left(\, \begin{bmatrix} x \\ y \end{bmatrix} \,\right)=\begin{bmatrix} x \\ -y \end{bmatrix}.$

Next, we prove that the function $F$ is a linear transformation from $\R^2$ to $\R^2$.
We need to verify the following two properties: for any $\mathbf{u}, \mathbf{v}\in \R^2$ and $c\in \R$, we have

1. $F(\mathbf{u}+\mathbf{v})=F(\mathbf{u})+F(\mathbf{v})$
2. $F(c\mathbf{u})=cF(\mathbf{u})$.

Let $\mathbf{u}=\begin{bmatrix} x \\ y \end{bmatrix}$ and $\mathbf{v}=\begin{bmatrix} x’ \\ y’ \end{bmatrix}$.
Then we have
\begin{align*}
F(\mathbf{u}+\mathbf{v})&=F\left(\, \begin{bmatrix}
x+x’ \\
y+y’

## Related Question.

The following problem is a generalization of the above problem.

Problem.
Let $T:\R^2 \to \R^2$ be a linear transformation of the $2$-dimensional vector space $\R^2$ (the $x$-$y$-plane) to itself of the reflection across a line $y=mx$ for some $m\in \R$.

Then find the matrix representation of the linear transformation $T$ with respect to the standard basis $B=\{\mathbf{e}_1, \mathbf{e}_2\}$ of $\R^2$, where
$\mathbf{e}_1=\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \mathbf{e}_2=\begin{bmatrix} 0 \\ 1 \end{bmatrix}.$

Note that if $m=0$, then this problem is the same as the current problem.

The solution is given in the post ↴
The Matrix for the Linear Transformation of the Reflection Across a Line in the Plane

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