Let $C=[0,1)\times [0,1)$. For any $v\in R^2$, we have $v=[v]+\{v\}$, where $[v]\in Z^2,\{v\}\in C$. $L$ is an euclidean segment of $C$ if it is the intersection of $C$ and a segment in an affine line of $R^2$.
For $A\in M_2(Z)$ being a integer matrix, let $T_A:C\rightarrow C,v\rightarrow\{Av\}$. And $v\in C$ is $T_A$-periodic if exists $n>0$ such that $T_A^nv=v$.
(1)Prove that if $\det(A)=0$, then $\operatorname{im}(A)$ is a finite union of euclidean segment of $C$.
(2)Proof that if $\det(A)\neq 0$, then the set of $T_A$-periodic points is dense in $C$.