Lösungen

1

a

a1

r = 1;
xVec0[t_] := r {Cos[t], Sin[t]};

ParametricPlot[xVec0[t], {t, 0, 2  Pi}, 
        AspectRatio -> Automatic];

a2

<< Graphics`ImplicitPlot`

fig0 = ({x, y} . {x, y} - r^2 == 0);

ImplicitPlot[fig0, {x, -1, 1}, 
        PlotStyle -> {GrayLevel[0], Dashing[{ .03}]}];

b

D1 = {{1, 0}, {0, 2}};  D1 // MatrixForm

b1

xVec1a[t_] := D1 . {Cos[t], Sin[t]};  xVec1a[t]

ParametricPlot[xVec1a[t], {t, 0, 2  Pi}, 
        AspectRatio -> Automatic];

b2

xVec1b[t_] := Inverse[D1] . {Cos[t], Sin[t]};  xVec1b[t]

ParametricPlot[xVec1b[t], {t, 0, 2  Pi}, 
        AspectRatio -> Automatic];

b3

(D1 . {x, y}) . (D1 . {x, y}) - r^2 == 0

fig1 = ((D1 . {x, y}) . (D1 . {x, y}) - r^2 == 
            0);

ImplicitPlot[fig1, {x, -1, 1}, 
        PlotStyle -> {GrayLevel[0], Dashing[{ .03}]}];

c

φ = -Pi/6;
M = {{Cos[φ], -Sin[φ]}, {Sin[φ], 
            Cos[φ]}}; 
M // MatrixForm

c1

xVec2a[t_] := M . xVec1a[t];  xVec2a[t]

ParametricPlot[xVec2a[t], {t, 0, 2  Pi}, 
        AspectRatio -> Automatic];

Eigensystem[M . D1]

Eigensystem[M . D1] // N

c2

xVec2b[t_] := Inverse[M] . xVec1b[t];  xVec2b[t]

ParametricPlot[xVec2b[t], {t, 0, 2  Pi}, 
        AspectRatio -> Automatic];

Eigensystem[Inverse[M] . Inverse[D1]]

Eigensystem[Inverse[M] . Inverse[D1]] // N

c3

fig2 = ((D1 . M . {x, y}) . (D1 . M . {x, y}) - r^2 == 
          0)

ImplicitPlot[fig2, {x, -1, 1}, 
        PlotStyle -> {GrayLevel[0], Dashing[{ .03}]}];

Eigensystem[D1 . M]

Eigensystem[D1 . M] // N