How would you represent a reflection over the x-axis using a transformation matrix?

To represent a reflection over the x-axis, you need to apply a transformation that effectively flips the coordinates of a point across the x-axis. This involves keeping the x-coordinate unchanged while inverting the y-coordinate.

The transformation matrix that achieves this for any point ((x, y)) can be represented as follows:

[

\begin{bmatrix}

1 & 0 \

0 & -1

\end{bmatrix}

]

When this matrix is applied to a point ((x, y)), the resulting point after the transformation can be computed as follows:

[

\begin{bmatrix}

1 & 0 \

0 & -1

\end{bmatrix}

\begin{bmatrix}

x \

y

\end{bmatrix}

=

\begin{bmatrix}

1 \cdot x + 0 \cdot y \

0 \cdot x + (-1) \cdot y

\end{bmatrix}

=

\begin{bmatrix}

x \

-y

\end{bmatrix}

]

This shows that the x-coordinate remains the same (since it multiplies by 1), and the y-coordinate changes its sign (as it