My book gives the following definition:
Let $L$ be a one dimensional subspace of $\mathbb R^2$. We may view $L$ as a line in the plane through the origin. A linear operator $T$ on $\mathbb R^2$ is called reflection of $\mathbb R^2$ about line $L$ if $T(x)=x$ for all $x \in L$ and $T(x)=-x$ for all $x \in L^\perp$.
I don't follow this definition at all, what is it trying to convey ? What is the difference between reflection about a line in $\mathbb R^2$ and reflection of $\mathbb R^2$ itself, about a line ? I am really confused.
Also what does $T(x)=x$ for all $x \in L$ and $T(x)=-x$ for all $x \in L^\perp$ mean ?
It would be nice it someone could explain this definition to me.
Also i am not very sure of what $L^\perp$ represents ? Is it just some sort of notation or does it mean the set of all those elements of $\mathbb R^2$ perpendicular to the line $L$?