一次変換[{{a11_,a12_},{a21_,a22_}}]:=Module[{A,f,g,b1,b2,F},
A={{a11,a12},{a21,a22}};
f[x_]:=A.x; b1=A.{1,0}; b2=A.{0,1};
y1=Table[Line[{k b1-b2,k b1+6b2}],{k,-1,6}];
y2=Table[Line[{-b1+k b2,k b2+6b1}],{k,-1,6}];
F={{0,0},{1,0},{1,2},{3,2},{3,3},{1,3},{1,4},{4,4},{4,5},{0,5},{0,0}};
原像={Pink,Thickness[0.008], Line[F]};
像={Blue,Thickness[0.008],Line[Map[f,F]]};
Show[Graphics[{原像, 像}],Axes->True,
GridLines->{Table[i,{i,-20,20}],Table[i,{i,-20,20}]},
PlotLabel->{{a11,a12},{a21,a22}},AspectRatio-> Automatic]]

一次変換[(({{-1, 0}, {0, 1}}))]

一次変換[(({{-3/5, 4/5}, {4/5, 3/5}}))]

M[θ_]=(({{Cos[θ], -Sin[θ]}, {Sin[θ], Cos[θ]}}))

一次変換[M[Pi/3]]

Show[Table[一次変換[MatrixPower[M[Pi/24],n]],{n,1,48,1}]]

Animate[一次変換[MatrixPower[M[Pi/12],n]],{n,1,24,1}]

M[a]//MatrixForm
M[b]//MatrixForm

M[a].M[a]//MatrixForm

M[a].M[a]//Simplify//MatrixForm

Solve[{y0+Y==3(x0+X),3(Y-y0)==-(X-x0)},{X,Y}]

一次変換[(({{-4/5, 3/5}, {3/5, 4/5}}))]