Lösungen

1

Remove["Global`*"]

a

X = Transpose[{{2, 1}, {-1, 2}}];  X // MatrixForm

Dλ = {{2, 0}, {0, 1}};  Dλ // MatrixForm

A = X . Dλ . Inverse[X];  A // MatrixForm

b

v1[t_] := {Cos[t], Sin[t]};
pl1 = 
        ParametricPlot[v1[t], {t, 0, 2  Pi}, 
          AspectRatio -> Automatic];

v2[t_] := A .  v1[t];   v2[t]

pl2 = 
        ParametricPlot[v2[t], {t, 0, 2  Pi}, 
          AspectRatio -> Automatic];

Show[pl1, pl2];

c

v3[t_] := {t, 1/2 t^2};  v3[t]

pl3 = 
        ParametricPlot[v3[t], {t, -2, 2}, 
          AspectRatio -> Automatic];

Dreh[φ_] := {{Cos[φ], -Sin[φ]}, {Sin[φ], Cos[φ]}};  
    Dreh[φ] // MatrixForm

φ = ArcTan[1/2]

v4[t_] := Dreh[φ] . v3[t] ; v4[
      t]

pl4 = 
        ParametricPlot[v4[t], {t, -2, 2}, 
          AspectRatio -> Automatic];

v5[t_] := A . v4[t];  v5[t]

pl5 = 
        ParametricPlot[v5[t], {t, -2, 2}, 
          AspectRatio -> Automatic];

Show[pl3, pl4];

Show[pl4, pl5];

2

Remove["Global`*"]

a

X[φ_] := 
      Transpose[{{Cos[φ], 
            Sin[φ]}, {-Sin[φ], Cos[φ]}}];  
    X[φ] // MatrixForm

λ2 = 1.5;
Dλ[λ_] := {{λ, 0}, {0, λ2}};  
    Dλ[λ] // MatrixForm

A[φ_, λ_] := 
      X[φ] . Dλ[λ] . Inverse[X[φ]];  
    A[φ, λ] // MatrixForm

b

v1[t_] := {Cos[t], Sin[t]};
pl1 = 
        ParametricPlot[v1[t], {t, 0, 2  Pi}, 
          AspectRatio -> Automatic];

v2[t_, φ_, λ_] :=  
      A[φ, λ] .  v1[t];   v2[
      t, φ, λ]

pl2[ φ_, λ_] := 
        ParametricPlot[
          Evaluate[v2[t, φ, λ]], {t, 0, 2  Pi}, 
          AspectRatio -> Automatic];

pl2[ 2  Pi/9, 1 + Sqrt[ 2  Pi/9]];

Show[
        Table[pl2[ φ, 1 + Sqrt[ φ]], { φ, 
            2  Pi/9, 2  Pi, 2  Pi/9}]];

Die eingehüllte Kurve ist der zentrische Kreis mit dem Radius  lambda2 = 1.5.