** Linear Transformation **

Clear[f];  t = Pi/6;
a1 = Cos[t]; b1 = -Sin[t]; a2 = Sin[t]; 
    b2 = Cos[t];
f[{x_, y_}] := {a1*x + b1*y, 
          a2*x + b2*y};
co = {}; curve = {};
Do[AppendTo[co, {{-5, y}, {5, y}}], {y, -5, 5}];
Do[AppendTo[co, {{x, -5}, {x, 5}}], {x, -5, 5}];
curve = 
        Table[{1.5  Sin[4  t], -3 Sin[3  t]*Sin[5  t] + 2}, {t, 0, 2  Pi, 
            Pi/180}];
Do[
        AppendTo[curve, {5  Sin[2  t], 5  Sin[3  t]*Sin[5  t]}], {t, 0, Pi, 
          Pi/180}];
Show[Graphics[Table[Line[co[[i]]], {i, 1, 22}]], 
      Graphics[Line[curve]], 
      AspectRatio -> Automatic]
w = 1;
Show[
      Graphics[Table[
          Line[{Nest[f, co[[i, 1]], w], Nest[f, co[[i, 2]], w]}], {i, 
            1, 22}]], 
      Graphics[Line[
          Table[Nest[f, curve[[j]], w], {j, 1, Length[curve]}]]], 
      AspectRatio -> Automatic]

Clear[f];
a1 = 1.1; b1 = 0.3; a2 = 0.2; b2 = 0.9;
f[{x_, y_}] := {a1*x + b1*y, 
          a2*x + b2*y};
w = 1;
Show[
      Graphics[Table[
          Line[{Nest[f, co[[i, 1]], w], Nest[f, co[[i, 2]], w]}], {i, 
            1, 22}]], 
      Graphics[Line[
          Table[Nest[f, curve[[j]], w], {j, 1, Length[curve]}]]], 
      AspectRatio -> Automatic]

pic={};n=90;
Do[p0={Cos[2m*Pi/n],Sin[2m*Pi/n]};
    AppendTo[pic,Line[{{0,0},2p0}]];
    points={};p=p0;Do[AppendTo[points,p];p1=f[p];p=p1,{k,1,2}];
    AppendTo[pic,Line[points]],
    {m,1,n}];
pic1=Show[Graphics[pic],AspectRatio->Automatic]

co = {}; curve = {};
Do[
        AppendTo[
          co, {{-5 - 0.8229  y, -5*0.5486 + y}, {5 - 0.8229  y, 
              5*0.5486 + y}}], {y, -5, 5}];
Do[
        AppendTo[
          co, {{x + 5*0.8229, 0.5486  x - 5}, {x - 5*0.8229, 
              0.5486  x + 5}}], {x, -5, 5}];
curve = 
        Table[{1.5  Sin[4  t], -3 Sin[3  t]*Sin[5  t] + 2}, {t, 0, 2  Pi, 
            Pi/180}];
Do[
        AppendTo[curve, {5  Sin[2  t], 5  Sin[3  t]*Sin[5  t]}], {t, 0, Pi, 
          Pi/180}];
Show[Graphics[Table[Line[co[[i]]], {i, 1, 22}]], 
      Graphics[Line[curve]], 
      AspectRatio -> Automatic]
w = 1;
Show[
      Graphics[Table[
          Line[{Nest[f, co[[i, 1]], w], Nest[f, co[[i, 2]], w]}], {i, 
            1, 22}]], 
      Graphics[Line[
          Table[Nest[f, curve[[j]], w], {j, 1, Length[curve]}]]], 
      AspectRatio -> Automatic]

pic={};
Do[x=2Random[]-1;y=2Random[]-1;s={x,y};
    Do[AppendTo[pic,s];s1=f[s];s=s1   
    ,{h,1,7}]
,{m,1,200}];
ListPlot[pic,AspectRatio->Automatic]

\(A = {{1, 0.5, 0.4}, {2, 1, 0.7}, {2.5, 1.0/0.7, 2}}; 
      B = {1, 1, 1};

      Do[B1 = A . B; c = Apply[Plus, B1]; B2 = B1/c; Print[B2]; 
        B = B2, \ 
\ \ {n, 1, 9}]\)