I'm studying geometry. We're classifying affinities, and my professor wrote: This affine transformation is an hyperbolic transformation. It has a fixed point and two invariant lines. The fixed point is inside the invariant lines, inside both of them.
My question is: Why the fixed points are inside invariant lines? Is this fact true with planes and invariant lines? Is this general or it only happens on this particular case?
Thanks!
Let $a,b\in\mathbb{R}^n$ such that $f(a)=a$ and $f(b)=b$. We know $f$ is of the form $f(x)=Ax+v$ for some matrix $A$ and vector $v$.
Moreover, the line joining $a$ and $b$ is $\{ta+(1-t)b|t\in\mathbb{R}\}$. Then
$\begin{eqnarray*}f(ta+(1-t)b)&=&A(ta+(1-t)b)+v\\ &=&tAa+(1-t)Ab+v\\ &=&tAa+tv\\ &&+(1-t)Ab+(1-t)v\\ &&-tv-(1-t)v+v\\ &=&tf(a)+(1-t)f(b)\\ &=&ta+(1-t)b \end{eqnarray*}$
Then, all the line is $f$-invariant.