如图，矩形$ABCD$的顶点坐标分别为$A(1,1)$，$B(2,1)$，$C(2,3)$，$D(1,3).$
$(1)$将矩形各顶点的横、纵坐标都乘以$2$，写出各对应点$A_{1}$、$B_{1}$、$C_{1}$、$D_{1}$的坐标；顺次连接$A_{1}B_{1}C_{1}D_{1}$，画出相应的图形．
$(2)$求矩形$A_{1}B_{1}C_{1}D_{1}$与矩形$ABCD$的面积的比 ______.