Definition of symmetry in $\mathbb{R}^n$ Are these two definitions of affine symmetry equivalent?


*

*A mapping $f: \mathbb R^n \rightarrow \mathbb R^n$ is called a symmetry with respect to an affine subspace $A\subset \mathbb R^n$ if $f$ is isometry (with respect to Euclidean metric), affine mapping, involution ($f\circ f=Id$) and the set of all fixed point  of $f$ is $A$.

*A mapping $f: \mathbb R^n \rightarrow \mathbb R^n$ is called symmetry with respect to an affine subspace $A\subset \mathbb R^n$ if 
$$
f(x)=x+2(p(x)-x) , x\in \mathbb R^n,
$$
where $p: \mathbb R^n \rightarrow \mathbb R^n$ is a orthogonal projection on $A$.

The first definition includes rotations and reflections whereas the second only has reflections.  e.g. In $\mathbb{R}^2$ consider the rotation of angle $\pi$.  It is an isometric, linear mapping that is an involution.  However, there is no way to write it as mapping in the second definition.  The set of fixed points of that rotation is the origin which forces $p(x) = 0$.  Then we are left with 
$$f(x) = x - 2x = -x$$
