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Find a matrix whose associated linear map maps the unit square to the shown parallelogram. Match figures \"of linear maps\" with matrices.

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Jede bijektive lineare Abbildung bildet das Einheitsquadrat auf ein Parallelogramm ab (vgl. Bemerkung 7.3 im Skript). Dabei werden die Standardbasisvektoren $e_1$ und $e_2$ auf die beiden Ecken des Parallelogramms abgebildet, die mit dem Ursprung durch eine Kante verbunden sind.

Aus der Zeichnung lesen wir ab, dass $\\mathbf f_A$ die Standardbasisvektoren $e_1$ und $e_2$ auf die Vektoren $\\var{A * vector(1,0)}$ und $\\var{A * vector(0,1)}$ abbilden muss (und jede lineare Abbildung, die das tut, bildet dann auch das Einheitsquadrat auf das grüne Parallelogramm ab). Andererseits ist $A e_1$ die erste und $A e_2$ die zweite Spalte von $A$. Die beiden möglichen Antworten auf die Frage sind daher

\\[ \\var{A}\\quad\\text{und}\\quad \\var{Ax}. \\]

Die Matrix $\\var{M[0]}$ ist eine Streckung, und zwar um den Faktor $\\var{M[0][0][0]}$ in Richtung der $x$-Achse und den Faktor $\\var{M[0][1][1]}$ in Richtung der $y$-Achse.

Die Spalten der Matrix $\\var{M[1]}$ sind linear abhängig, daher ist das Bild der zugehörigen linearen Abbildung eine Ursprungsgerade.

Die Matrix $\\var{M[2]}$ ist eine Scherung.

Die Matrix $\\var[fractionNumbers]{M[3]}$ ist näherungsweise eine Drehung um den Ursprung. Siehe Ergänzung 7.59 im Skript.

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Swap columns of $A$, this is the other acceptable solution in part a).

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Wir betrachten die folgende Zeichnung:

{plot(A)}

Geben Sie eine Matrix $A\\in M_2(\\mathbb R)$ an, so dass die Abbildung $\\mathbf f_A\\colon \\mathbb R^2\\to\\mathbb R^2$, $x\\mapsto Ax$ das \"Einheitsquadrat\" mit den Eckpunkten $\\begin{pmatrix} 0 \\\\ 0\\end{pmatrix}$, $\\begin{pmatrix} 1 \\\\ 0\\end{pmatrix}$, $\\begin{pmatrix} 0 \\\\ 1\\end{pmatrix}$, $\\begin{pmatrix} 1 \\\\ 1\\end{pmatrix}$ auf das im Bild gezeigte grüne Parallelogramm abbildet.

[[0]]

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Überlegen Sie sich, wie Sie an dem Parallelogramm die Wert $\\mathbf f_A(e_1)$, $\\mathbf f_A(e_2)$ der Standardbasisvektoren ablesen können.
Was sagen diese Werte über die Matrix $A$?

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Wir betrachten die folgenden 4 Skizzen:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

Skizze 1	Skizze 2	Skizze 3	Skizze 4
{plot1(Mshuffled[0])}	{plot1(Mshuffled[1])}	{plot1(Mshuffled[2])}	{plot1(Mshuffled[3])}

In jeder Skizze ist das grüne Parallelogramm das Bild des blauen Quadrats unter einer Abbildung der Form $\\mathbf f_M\\colon \\mathbb R^2\\to \\mathbb R^2$, $x\\mapsto Mx$. Ordnen Sie die 4 Skizzen oben den folgenden Matrizen zu:

\\[ A = \\var[fractionnumbers]{M[0]},\\quad B = \\var[fractionnumbers]{M[1]},\\quad C = \\var[fractionnumbers]{M[2]},\\quad D = \\var[fractionnumbers]{M[3]}\\]

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