Question

For a vector $\bold{v}$, let $\bold{r}$ be the reflection of $\bold{v}$ over the line
\[\bold{x} = t \begin{pmatrix} 2 \\ -1 \end{pmatrix}.\]

[asy]
unitsize(1 cm);

pair O, V, R;

O = (0,0);
V = (3,1);
R = reflect(O,(2,-1))*(V);

xaxis(-2,4);
yaxis(-3,1);

draw((-1)*(2,-1)--2*(2,-1));
draw(O--V,Arrow(8));
draw(O--R,Arrow(8));

dot(O);

label("$\mathbf{v}$", (O + V)/2, N);
label("$\mathbf{r}$", (O + R)/2, W);
[/asy]

There exists a $2 \times 2$ matrix $\bold{R}$ such that
\[\bold{r} = \bold{R} \bold{v}\]
for all 2-dimensional vectors $\bold{v}$. Find $\bold{R}$.