Two quadrics $Q_1$ and $Q_2$ in $\mathbb{R}^n$ are affine equivalent, $Q_1\sim Q_2$, if there exists an affine map $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ with $Q_2=f(Q_1)$.
How do I prove, that $\sim$ is an equivalence relation on the set of all quadrics in $\mathbb{R}^n$?
For reflexivity we have $Id_n$, so $Q_1=Id_n(Q_1)$.
For symmetry we have $f^{-1}$, so if $Q_2=f(Q_1)$ then $f^{-1}(Q_2)=Q_1$.
But what about transitivity? I'm also struggling with writing it down formally. Or are the given examples enough for a proof?
Since my score is not high enough to write a comment, I will just put a solution. Just one fact: The composition of two affine maps is still an affine map, I believe?