linAbb[a_,b_,c_,d_]:=
Module[{m={{a,b},{c,d}},
            mm,
           quadrat =   {{1,1},{-1,1},{-1,-1},{1,-1},{1,1}},
           raute      =   {{1,0},{0,1},{-1,0},{0,-1},{1,0}},
           deckel    =   {{w=(Sqrt[2.]),0},  {1,1},{0,w},{-1,1},{-w,0},{-1,-1}},
           kreis      =    Table[ { Cos[t],             Sin[t]},  {t,0,2 π,(π/30)}],
            ellipse =   Table[ {2 Cos[t],1.157 Sin[t]},{t,0,2 π,(π/30)}]
            },

                 mm[{x_,y_}]:=m.{x,y};

               Show[Graphics[ {AbsoluteThickness[2],
                Green, Line[ Map[mm,quadrat] ],
                Red,     Line[ Map[mm,raute] ],
                Blue,   Line[ Map[mm,deckel] ],
                Black, Line[ Map[mm,kreis] ],
                Brown, Line[ Map[mm,ellipse] ]
                                       } ],
                     AspectRatio->Automatic,Axes->True,  Frame->True,
                     PlotRange->1.5{{-2,2},{-2,2}},ImageSize->{500,500}]
     ];

Manipulate[
Column[{
Text@Row[{MatrixForm[{{"a","b"},{"c","d"}}],
" = ",
RawBoxes@"(",Grid[Map[NumberForm[#,{5,2},NumberSigns->{"-","+"}]&,{{a,b},{c,d}},{2}],Alignment->Right],RawBoxes@")"}],


linAbb[a,b,c,d]
          },Center],

{{a,1,"a"},-1,1,ControlPlacement->Right},
{{b,0,"b"},-1,1,ControlPlacement->Right},
{{c,0,"c"},-1,1,ControlPlacement->Right},
{{d,1,"d"},-1,1,ControlPlacement->Right}

]