LineareAlgebra.Aufgaben01 History
Show minor edits - Show changes to output
Changed lines 25-26 from:
'''Aufgabe 4''' Gegeben sind folgende Matrizen (bzw{.} Vektoren):
to:
'''Aufgabe 4''' Gegeben sind folgende Matrizen:
Changed lines 54-58 from:
3x-y+z+u & = & -7 \\
x+2y-z+2u & = & -4\\
x-y-z-u & = & -1 \\
2x+y+2z+u & =&-1
\end{eqnarray*}
x+2y-z+2u & = & -4\\
x-y-z-u & = & -1 \\
2x+y+2z+u & =&-1
\end{eqnarray*}
to:
3x-y+z+u & = & -7 \\ x+2y-z+2u & = & -4\\ x-y-z-u & = & -1 \\ 2x+y+2z+u & =&-1 \end{eqnarray*}
Changed lines 51-52 from:
to:
'''Aufgabe 6''' Löse das folgende Gleichungssystem mit Hilfe des Gauß-Algorithmus.
$$
\begin{eqnarray*}
3x-y+z+u & = & -7 \\
x+2y-z+2u & = & -4\\
x-y-z-u & = & -1 \\
2x+y+2z+u & =&-1
\end{eqnarray*}
$$
$$
\begin{eqnarray*}
3x-y+z+u & = & -7 \\
x+2y-z+2u & = & -4\\
x-y-z-u & = & -1 \\
2x+y+2z+u & =&-1
\end{eqnarray*}
$$
Changed lines 41-44 from:
2x+4z & = & 1 + y \\
2y & = & 2+6z
\end{array}
\end{array}
to:
2x+4z & = & 1 + y \\ -3x+2y & = & 3\\ 2y & = & 2+6z \end{array}
Changed lines 38-55 from:
to:
'''Aufgabe 5''' Stelle das lineare Gleichungssystem in Matrixschreibweise dar und löse es. Die benötigte inverse Matrix findet sich unter den vier angegebenen.
$$
\begin{array}{rcl}
2x+4z & = & 1 + y \\
-3x+2y & = & 3\\
2y & = & 2+6z
\end{array}
$$
$$
\scriptsize\frac{1}{30}\cdot\left(\begin{array}{ccc}8 &-2 & 8\\ 18 & 12 & 12 \\ 6 & 2 & -1\end{array}\right)\qquad\quad
\frac{1}{30}\cdot\left(\begin{array}{ccc}12 &-2 & 8\\ 6 & 4 & 4 \\ 6 & 4 & -1\end{array}\right)\qquad\quad
\frac{1}{30}\cdot\left(\begin{array}{ccc}12 &-2 & 8\\ 18 & 12 & 12 \\ 6 & 4 & -1\end{array}\right)\qquad\quad
\frac{1}{30}\cdot\left(\begin{array}{ccc}12 &-2 & -8\\ 8 & 12 & 12 \\ 6 & 4 & -1\end{array}\right)
$$
$$
\begin{array}{rcl}
2x+4z & = & 1 + y \\
-3x+2y & = & 3\\
2y & = & 2+6z
\end{array}
$$
$$
\scriptsize\frac{1}{30}\cdot\left(\begin{array}{ccc}8 &-2 & 8\\ 18 & 12 & 12 \\ 6 & 2 & -1\end{array}\right)\qquad\quad
\frac{1}{30}\cdot\left(\begin{array}{ccc}12 &-2 & 8\\ 6 & 4 & 4 \\ 6 & 4 & -1\end{array}\right)\qquad\quad
\frac{1}{30}\cdot\left(\begin{array}{ccc}12 &-2 & 8\\ 18 & 12 & 12 \\ 6 & 4 & -1\end{array}\right)\qquad\quad
\frac{1}{30}\cdot\left(\begin{array}{ccc}12 &-2 & -8\\ 8 & 12 & 12 \\ 6 & 4 & -1\end{array}\right)
$$
Changed lines 32-34 from:
E=\left(\begin{array}{cc}-1 &-4 \\ 3 & 2 \end{array}\right),\quad
F=\left(\begin{array}{c}-1 \\ 2 \\ 3 \end{array}\right),\quad
G=\left(\begin{array}{cc}5 & 7 \end{array}\right)
to:
E=\left(\begin{array}{cc}-1 &-4 \\ 3 & 2 \end{array}\right),\quad F=\left(\begin{array}{c}-1 \\ 2 \\ 3 \end{array}\right),\quad G=\left(\begin{array}{cc}5 & 7 \end{array}\right)
Added line 34:
Added line 26:
Changed lines 28-31 from:
A=\left(\begin{array}{cc}2 &3 \\ -5 & 1 \end{array}\right),\quad
B=\left(\begin{array}{c}-3 \\-2\end{array}\right),\quad
C=\left(\begin{array}{ccc}-1 &4 &-2 \\ 2 & 3&5 \end{array}\right),\quad
D=\left(\begin{array}{cc}3 &1 \\ -2 & 4 \\1&-3 \end{array}\right)
to:
A=\left(\begin{array}{cc}2 &3 \\ -5 & 1 \end{array}\right),\quad B=\left(\begin{array}{c}-3 \\-2\end{array}\right),\quad C=\left(\begin{array}{ccc}-1 &4 &-2 \\ 2 & 3&5 \end{array}\right),\quad D=\left(\begin{array}{cc}3 &1 \\ -2 & 4 \\1&-3 \end{array}\right)
Added line 30:
Added lines 25-41:
'''Aufgabe 4''' Gegeben sind folgende Matrizen (bzw{.} Vektoren):
$$
A=\left(\begin{array}{cc}2 &3 \\ -5 & 1 \end{array}\right),\quad
B=\left(\begin{array}{c}-3 \\-2\end{array}\right),\quad
C=\left(\begin{array}{ccc}-1 &4 &-2 \\ 2 & 3&5 \end{array}\right),\quad
D=\left(\begin{array}{cc}3 &1 \\ -2 & 4 \\1&-3 \end{array}\right)
$$
$$
E=\left(\begin{array}{cc}-1 &-4 \\ 3 & 2 \end{array}\right),\quad
F=\left(\begin{array}{c}-1 \\ 2 \\ 3 \end{array}\right),\quad
G=\left(\begin{array}{cc}5 & 7 \end{array}\right)
$$
#Welche der angegebenen Produkte lassen sich bilden? Warum? {$AB,BA,AE,EA,CD,DC,GA,GB,BG,FC,CF,ED,DE$}
#Betrachte nur {$B$}, {$C$} und {$G$}. Berechne untereinander alle möglichen Produkte.
#Formuliere eine Bedingung dafür, wann zwei Matrizen miteinander multipliziert werden können.
$$
A=\left(\begin{array}{cc}2 &3 \\ -5 & 1 \end{array}\right),\quad
B=\left(\begin{array}{c}-3 \\-2\end{array}\right),\quad
C=\left(\begin{array}{ccc}-1 &4 &-2 \\ 2 & 3&5 \end{array}\right),\quad
D=\left(\begin{array}{cc}3 &1 \\ -2 & 4 \\1&-3 \end{array}\right)
$$
$$
E=\left(\begin{array}{cc}-1 &-4 \\ 3 & 2 \end{array}\right),\quad
F=\left(\begin{array}{c}-1 \\ 2 \\ 3 \end{array}\right),\quad
G=\left(\begin{array}{cc}5 & 7 \end{array}\right)
$$
#Welche der angegebenen Produkte lassen sich bilden? Warum? {$AB,BA,AE,EA,CD,DC,GA,GB,BG,FC,CF,ED,DE$}
#Betrachte nur {$B$}, {$C$} und {$G$}. Berechne untereinander alle möglichen Produkte.
#Formuliere eine Bedingung dafür, wann zwei Matrizen miteinander multipliziert werden können.
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to:
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Changed line 21 from:
\mbox{a)}\; A= \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0&0&3\end{array} \right) \quad \mbox{b)}\; A= \left( \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 2 & 0 \\ 3&0&0\end{array} \right) \quad \mbox{c)}\; A= \left( \begin{array}{cc} a & 1 \\ 0 & b\end{array} \right) \quad \mbox{d)}\; A= \left( \begin{array}{cc} a & 0 \\ 1 & b\end{array} \right) \quad \mbox{e)}\; A= \left( \begin{array}{ccc} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0&0&1\end{array} \right)
to:
\mbox{a)}\; A= \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0&0&3\end{array} \right) \quad \mbox{b)}\; A= \left( \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 2 & 0 \\ 3&0&0\end{array} \right) \quad \mbox{c)}\; A= \left( \begin{array}{cc} a & 1 \\ 0 & b\end{array} \right) \quad \mbox{d)}\; A= \left( \begin{array}{ccc} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0&0&1\end{array} \right)
Changed line 21 from:
\mbox{a)}\; A= \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0&0&3\end{array} \right) \quad \mbox{b)}\; A= \left( \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 2 & 0 \\ 3&0&0\end{array} \right) \quad \mbox{c)}\; A= \left( \begin{array}{cc} a & 1 \\ 0 & b\end{array} \right) \quad \mbox{d)}\; A= \left( \begin{array}{cc} a & 0 \\ 1 & b\end{array} \right) \quad \mbox{e)}\; A= \left( \begin{array}{ccc} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0&0&1\end{array}
to:
\mbox{a)}\; A= \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0&0&3\end{array} \right) \quad \mbox{b)}\; A= \left( \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 2 & 0 \\ 3&0&0\end{array} \right) \quad \mbox{c)}\; A= \left( \begin{array}{cc} a & 1 \\ 0 & b\end{array} \right) \quad \mbox{d)}\; A= \left( \begin{array}{cc} a & 0 \\ 1 & b\end{array} \right) \quad \mbox{e)}\; A= \left( \begin{array}{ccc} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0&0&1\end{array} \right)
Changed line 21 from:
\mbox{a)}\; A= \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0&0&3\end{array} \right) \quad \mbox{b)}\; A= \left( \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 2 & 0 \\ 3&0&0\end{array} \right) \quad \mbox{c)}\; A= \left( \begin{array}{cc} a & 1 \\ 0 & b\end{array} \right)
to:
\mbox{a)}\; A= \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0&0&3\end{array} \right) \quad \mbox{b)}\; A= \left( \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 2 & 0 \\ 3&0&0\end{array} \right) \quad \mbox{c)}\; A= \left( \begin{array}{cc} a & 1 \\ 0 & b\end{array} \right) \quad \mbox{d)}\; A= \left( \begin{array}{cc} a & 0 \\ 1 & b\end{array} \right) \quad \mbox{e)}\; A= \left( \begin{array}{ccc} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0&0&1\end{array}
Changed lines 17-22 from:
$$
to:
$$
'''Aufgabe 3''' Berechne jeweils {$A^2$}, {$A^3$} und {$A^4$}.
$$
\mbox{a)}\; A= \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0&0&3\end{array} \right) \quad \mbox{b)}\; A= \left( \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 2 & 0 \\ 3&0&0\end{array} \right) \quad \mbox{c)}\; A= \left( \begin{array}{cc} a & 1 \\ 0 & b\end{array} \right)
$$
'''Aufgabe 3''' Berechne jeweils {$A^2$}, {$A^3$} und {$A^4$}.
$$
\mbox{a)}\; A= \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0&0&3\end{array} \right) \quad \mbox{b)}\; A= \left( \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 2 & 0 \\ 3&0&0\end{array} \right) \quad \mbox{c)}\; A= \left( \begin{array}{cc} a & 1 \\ 0 & b\end{array} \right)
$$
Changed line 8 from:
\mbox{d)}\; & \left( \begin{array}{ccc} 2 & -1 & 12 \end{array} \right)\cdot \left( \begin{array}{c} 6\\ 0 \\ -3 \end{array} \right) \quad & \mbox{e)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0,5 & 0,3 \end{array} \right)\cdot \left( \begin{array}{c} 5\\ 5 \end{array} \right) \quad & \mbox{c)}\; & \left( \begin{array}{cc} 7 & -7\3 & 2 \end{array} \right)\cdot \left( \begin{array}{c} -2\2 \end{array} \right) \end{array}
to:
\mbox{d)}\; & \left( \begin{array}{ccc} 2 & -1 & 12 \end{array} \right)\cdot \left( \begin{array}{c} 6\\ 0 \\ -3 \end{array} \right) \quad & \mbox{e)}\; & \left( \begin{array}{cc} 0,3 & -2 \\ 0,7 & 0,1 \end{array} \right)\cdot \left( \begin{array}{c} 10\\ 5 \end{array} \right) \quad & \mbox{f)}\; & \left( \begin{array}{cc} \sqrt{2} & -\sqrt{3}\\ -\sqrt{2} & \sqrt{3} \end{array} \right)\cdot \left( \begin{array}{c} \sqrt{2}\\ \sqrt{3} \end{array} \right) \end{array}
Changed lines 7-8 from:
\mbox{a)}\; & \left( \begin{array}{ccc} -3 & 1 & 8 \\ 5 & 7 & 0 \\ 0 & -2 & 3 \end{array} \right)\cdot \left( \begin{array}{c} 1\\ 2 \\ 3 \end{array} \right) \quad & \mbox{b)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0,5 & 0,3 \end{array} \right)\cdot \left( \begin{array}{c} 5\\ 5 \end{array} \right) \quad & \mbox{c)}\; & \left( \begin{array}{cc} 7 & -7\\ 3 & 2 \end{array} \right)\cdot \left( \begin{array}{c} -2\\ 2 \end{array} \right) \\ \mbox{} \\
\mbox{a)}\; & \left( \begin{array}{ccc} -3 & 1 & 8 \ 5 & 7 & 0 \& -2 & 3 \end{array} \right)\cdot \left( \begin{array}{c} 1\\ 2 \\ 3 \end{array} \right) \quad & \mbox{b)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0,5 & 0,3 \end{array} \right)\cdot \left( \begin{array}{c} 5\\ 5 \end{array} \right) \quad & \mbox{c)}\; & \left( \begin{array}{cc} 7 & -7\\ 3 & 2 \end{array} \right)\cdot \left( \begin{array}{c} -2\\ 2 \end{array} \right) \end{array}
\mbox
to:
\mbox{a)}\; & \left( \begin{array}{ccc} -3 & 1 & 8 \\ 5 & 7 & 0 \\ 0 & -2 & 3 \end{array} \right)\cdot \left( \begin{array}{c} 1\\ 2 \\ 3 \end{array} \right) \quad & \mbox{b)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0,5 & 0,3 \end{array} \right)\cdot \left( \begin{array}{c} 5\\ 5 \end{array} \right) \quad & \mbox{c)}\; & \left( \begin{array}{cc} 7 & -7\\ 3 & 2 \end{array} \right)\cdot \left( \begin{array}{c} -2\\ 2 \end{array} \right) \\
\mbox{d)}\; & \left( \begin{array}{ccc} 2 & -1 & 12 \end{array} \right)\cdot \left( \begin{array}{c} 6\\ 0 \\ -3 \end{array} \right) \quad & \mbox{e)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0,5 & 0,3 \end{array} \right)\cdot \left( \begin{array}{c} 5\\ 5 \end{array} \right) \quad & \mbox{c)}\; & \left( \begin{array}{cc} 7 & -7\\ 3 & 2 \end{array} \right)\cdot \left( \begin{array}{c} -2\\ 2 \end{array} \right) \end{array}
\mbox{d)}\; & \left( \begin{array}{ccc} 2 & -1 & 12 \end{array} \right)\cdot \left( \begin{array}{c} 6\\ 0 \\ -3 \end{array} \right) \quad & \mbox{e)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0,5 & 0,3 \end{array} \right)\cdot \left( \begin{array}{c} 5\\ 5 \end{array} \right) \quad & \mbox{c)}\; & \left( \begin{array}{cc} 7 & -7\\ 3 & 2 \end{array} \right)\cdot \left( \begin{array}{c} -2\\ 2 \end{array} \right) \end{array}
Changed line 7 from:
\mbox{a)}\; & \left( \begin{array}{ccc} -3 & 1 & 8 \\ 5 & 7 & 0 \\ 0 & -2 & 3 \end{array} \right)\cdot \left( \begin{array}{c} 1\\ 2 \\ 3 \end{array} \right) \quad & \mbox{b)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0,5 & 0,3 \end{array} \right)\cdot \left( \begin{array}{c} 5\\ 5 \end{array} \right) \quad & \mbox{c)}\; & \left( \begin{array}{cc} 7 & -7\\ 3 & 2 \end{array} \right)\cdot \left( \begin{array}{c} -2\\ 2 \end{array} \right) \\\vspace{1em} \\
to:
\mbox{a)}\; & \left( \begin{array}{ccc} -3 & 1 & 8 \\ 5 & 7 & 0 \\ 0 & -2 & 3 \end{array} \right)\cdot \left( \begin{array}{c} 1\\ 2 \\ 3 \end{array} \right) \quad & \mbox{b)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0,5 & 0,3 \end{array} \right)\cdot \left( \begin{array}{c} 5\\ 5 \end{array} \right) \quad & \mbox{c)}\; & \left( \begin{array}{cc} 7 & -7\\ 3 & 2 \end{array} \right)\cdot \left( \begin{array}{c} -2\\ 2 \end{array} \right) \\ \mbox{ } \\
Changed line 7 from:
\mbox{a)}\; & \left( \begin{array}{ccc} -3 & 1 & 8 \\ 5 & 7 & 0 \\ 0 & -2 & 3 \end{array} \right)\cdot \left( \begin{array}{c} 1\\ 2 \\ 3 \end{array} \right) \quad & \mbox{b)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0,5 & 0,3 \end{array} \right)\cdot \left( \begin{array}{c} 5\\ 5 \end{array} \right) \quad & \mbox{c)}\; & \left( \begin{array}{cc} 7 & -7\\ 3 & 2 \end{array} \right)\cdot \left( \begin{array}{c} -2\\ 2 \end{array} \right) \\ \\
to:
\mbox{a)}\; & \left( \begin{array}{ccc} -3 & 1 & 8 \\ 5 & 7 & 0 \\ 0 & -2 & 3 \end{array} \right)\cdot \left( \begin{array}{c} 1\\ 2 \\ 3 \end{array} \right) \quad & \mbox{b)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0,5 & 0,3 \end{array} \right)\cdot \left( \begin{array}{c} 5\\ 5 \end{array} \right) \quad & \mbox{c)}\; & \left( \begin{array}{cc} 7 & -7\\ 3 & 2 \end{array} \right)\cdot \left( \begin{array}{c} -2\\ 2 \end{array} \right) \\\vspace{1em} \\
Changed line 7 from:
\mbox{a)}\; & \left( \begin{array}{ccc} -3 & 1 & 8 \\ 5 & 7 & 0 \\ 0 & -2 & 3 \end{array} \right)\cdot \left( \begin{array}{c} 1\\ 2 \\ 3 \end{array} \right) \quad & \mbox{b)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0,5 & 0,3 \end{array} \right)\cdot \left( \begin{array}{c} 5\\ 5 \end{array} \right) \quad & \mbox{c)}\; & \left( \begin{array}{cc} 7 & -7\\ 3 & 2 \end{array} \right)\cdot \left( \begin{array}{c} -2\\ 2 \end{array} \right) \\
to:
\mbox{a)}\; & \left( \begin{array}{ccc} -3 & 1 & 8 \\ 5 & 7 & 0 \\ 0 & -2 & 3 \end{array} \right)\cdot \left( \begin{array}{c} 1\\ 2 \\ 3 \end{array} \right) \quad & \mbox{b)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0,5 & 0,3 \end{array} \right)\cdot \left( \begin{array}{c} 5\\ 5 \end{array} \right) \quad & \mbox{c)}\; & \left( \begin{array}{cc} 7 & -7\\ 3 & 2 \end{array} \right)\cdot \left( \begin{array}{c} -2\\ 2 \end{array} \right) \\ \\
Changed line 7 from:
\mbox{a)}\; & \left( \begin{array}{ccc} -3 & 1 & 8 \\ 5 & 7 & 0 \\ 0 & -2 & 3 \end{array} \right)\cdot \left( \begin{array}{c} 1\\ 2 \\ 3 \end{array} \right) \quad & \mbox{b)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0,5 & 0,3 \end{array} \right)\cdot \left( \begin{array}{c} 5\\ 5 \end{array} \right) \quad & \mbox{c)}\; & \left( \begin{array}{cc} 7 & -7\\ 3 & 2 \end{array} \right)\cdot \left( \begin{array}{c} -2\\ 2 \end{array} \right) \\
to:
\mbox{a)}\; & \left( \begin{array}{ccc} -3 & 1 & 8 \\ 5 & 7 & 0 \\ 0 & -2 & 3 \end{array} \right)\cdot \left( \begin{array}{c} 1\\ 2 \\ 3 \end{array} \right) \quad & \mbox{b)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0,5 & 0,3 \end{array} \right)\cdot \left( \begin{array}{c} 5\\ 5 \end{array} \right) \quad & \mbox{c)}\; & \left( \begin{array}{cc} 7 & -7\\ 3 & 2 \end{array} \right)\cdot \left( \begin{array}{c} -2\\ 2 \end{array} \right) \\
Added line 3:
Changed lines 7-8 from:
\mbox{a)}\; & \left( \begin{array}{ccc} -3 & 1 & 8 \\ 5 & 7 & 0 \\ 0 & -2 & 3 \end{array} \right)\cdot \left( \begin{array}{c} 1\\ 2 \\ 3 \end{array} \right) \quad & \mbox{b)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0,5 & 0,3 \end{array} \right)\cdot \left( \begin{array}{c} 5\\ 5 \end{array} \right) \quad & \mbox{c)}\; & \left( \begin{array}{cc} 7 & -7\\ 3 & 2 \end{array} \right)\cdot \left( \begin{array}{c} -2\\ 2 \end{array} \right) \end{array}
to:
\mbox{a)}\; & \left( \begin{array}{ccc} -3 & 1 & 8 \\ 5 & 7 & 0 \\ 0 & -2 & 3 \end{array} \right)\cdot \left( \begin{array}{c} 1\\ 2 \\ 3 \end{array} \right) \quad & \mbox{b)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0,5 & 0,3 \end{array} \right)\cdot \left( \begin{array}{c} 5\\ 5 \end{array} \right) \quad & \mbox{c)}\; & \left( \begin{array}{cc} 7 & -7\\ 3 & 2 \end{array} \right)\cdot \left( \begin{array}{c} -2\\ 2 \end{array} \right) \\
\mbox{a)}\; & \left( \begin{array}{ccc} -3 & 1 & 8 \\ 5 & 7 & 0 \\ 0 & -2 & 3 \end{array} \right)\cdot \left( \begin{array}{c} 1\\ 2 \\ 3 \end{array} \right) \quad & \mbox{b)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0,5 & 0,3 \end{array} \right)\cdot \left( \begin{array}{c} 5\\ 5 \end{array} \right) \quad & \mbox{c)}\; & \left( \begin{array}{cc} 7 & -7\\ 3 & 2 \end{array} \right)\cdot \left( \begin{array}{c} -2\\ 2 \end{array} \right) \end{array}
\mbox{a)}\; & \left( \begin{array}{ccc} -3 & 1 & 8 \\ 5 & 7 & 0 \\ 0 & -2 & 3 \end{array} \right)\cdot \left( \begin{array}{c} 1\\ 2 \\ 3 \end{array} \right) \quad & \mbox{b)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0,5 & 0,3 \end{array} \right)\cdot \left( \begin{array}{c} 5\\ 5 \end{array} \right) \quad & \mbox{c)}\; & \left( \begin{array}{cc} 7 & -7\\ 3 & 2 \end{array} \right)\cdot \left( \begin{array}{c} -2\\ 2 \end{array} \right) \end{array}
Changed lines 5-6 from:
\begin{array}{llll}
\mbox{a)}\; & \left( \begin{array}{ccc} -3 & 1 & 8 \\ 5 & 7 & 0 \\ 0 & -2 & 3 \end{array} \right)\cdot \left( \begin{array}{c} 1\\ 2 \\ 3 \end{array} \right) \quad & \mbox{b)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0 & 0,5 & 0,3 \end{array} \right)\cdot \left( \begin{array}{c} 5\\ 5 \end{array} \right) \quad \end{array}
\mbox{a)}\; & \left( \begin{array}{ccc} -3 & 1 & 8 \\ 5 & 7 & 0 \\ 0 & -2 & 3 \end{array} \right)\cdot \left( \begin{array}{c} 1\\ 2 \\ 3 \end{array} \right) \quad & \mbox{b)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0 & 0,
to:
\begin{array}{llllll}
\mbox{a)}\; & \left( \begin{array}{ccc} -3 & 1 & 8 \\ 5 & 7 & 0 \\ 0 & -2 & 3 \end{array} \right)\cdot \left( \begin{array}{c} 1\\ 2 \\ 3 \end{array} \right) \quad & \mbox{b)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0,5 & 0,3 \end{array} \right)\cdot \left( \begin{array}{c} 5\\ 5 \end{array} \right) \quad & \mbox{c)}\; & \left( \begin{array}{cc} 7 & -7\\ 3 & 2 \end{array} \right)\cdot \left( \begin{array}{c} -2\\ 2 \end{array} \right) \end{array}
\mbox{a)}\; & \left( \begin{array}{ccc} -3 & 1 & 8 \\ 5 & 7 & 0 \\ 0 & -2 & 3 \end{array} \right)\cdot \left( \begin{array}{c} 1\\ 2 \\ 3 \end{array} \right) \quad & \mbox{b)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0,5 & 0,3 \end{array} \right)\cdot \left( \begin{array}{c} 5\\ 5 \end{array} \right) \quad & \mbox{c)}\; & \left( \begin{array}{cc} 7 & -7\\ 3 & 2 \end{array} \right)\cdot \left( \begin{array}{c} -2\\ 2 \end{array} \right) \end{array}
Changed line 6 from:
\mabox{a)}\; & \left( \begin{array}{ccc} -3 & 1 & 8 \\ 5 & 7 & 0 \\ 0 & -2 & 3 \end{array} \right)\cdot \left( \begin{array}{c} 1\\ 2 \\ 3 \end{array} \right) \quad & \mbox{b)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0 & 0,5 & 0,3 \end{array} \right)\cdot \left( \begin{array}{c} 5\\ 5 \end{array} \right) \quad \end{array}
to:
\mbox{a)}\; & \left( \begin{array}{ccc} -3 & 1 & 8 \\ 5 & 7 & 0 \\ 0 & -2 & 3 \end{array} \right)\cdot \left( \begin{array}{c} 1\\ 2 \\ 3 \end{array} \right) \quad & \mbox{b)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0 & 0,5 & 0,3 \end{array} \right)\cdot \left( \begin{array}{c} 5\\ 5 \end{array} \right) \quad \end{array}
Changed lines 1-9 from:
to:
!!Übungsaufgaben: Rechnen mit Matrizen
'''Aufgabe1''' Berechne.
$$
\begin{array}{llll}
\mabox{a)}\; & \left( \begin{array}{ccc} -3 & 1 & 8 \\ 5 & 7 & 0 \\ 0 & -2 & 3 \end{array} \right)\cdot \left( \begin{array}{c} 1\\ 2 \\ 3 \end{array} \right) \quad & \mbox{b)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0 & 0,5 & 0,3 \end{array} \right)\cdot \left( \begin{array}{c} 5\\ 5 \end{array} \right) \quad \end{array}
$$
'''Aufgabe 2''' Bilde für die folgenden Matrizen alle möglichen Matrixprodukte:
'''Aufgabe1''' Berechne.
$$
\begin{array}{llll}
\mabox{a)}\; & \left( \begin{array}{ccc} -3 & 1 & 8 \\ 5 & 7 & 0 \\ 0 & -2 & 3 \end{array} \right)\cdot \left( \begin{array}{c} 1\\ 2 \\ 3 \end{array} \right) \quad & \mbox{b)}\; & \left( \begin{array}{cc} 1 & -1 \\ 3 & 0 \\ 0 & 0,5 & 0,3 \end{array} \right)\cdot \left( \begin{array}{c} 5\\ 5 \end{array} \right) \quad \end{array}
$$
'''Aufgabe 2''' Bilde für die folgenden Matrizen alle möglichen Matrixprodukte:
Changed line 1 from:
Bilde für die folgenden Matrizen alle möglichen Matrixprodukte:
to:
'''Aufgabe 1''' Bilde für die folgenden Matrizen alle möglichen Matrixprodukte:
Changed lines 2-5 from:
$$ A= \left( \begin{array}{ccc} 1 & 4 & -2 \\
3 & 5 & 0 \end{array} \right) \; ,\;
B= \left( \begin{array}{cc} 3 & -1 \\
0 & 2 \end{array} \right) \; ,\;
B= \left( \begin{array}{cc} 3 & -1 \\
0 & 2 \end{array} \right) \; ,\;
to:
$$ A= \left( \begin{array}{ccc} 1 & 4 & -2 \\ 3 & 5 & 0 \end{array} \right) \; ,\;
B= \left( \begin{array}{cc} 3 & -1 \\ 0 & 2 \end{array} \right) \; ,\;
B= \left( \begin{array}{cc} 3 & -1 \\ 0 & 2 \end{array} \right) \; ,\;
Changed lines 2-3 from:
$$
A= \left( \begin{array}{ccc} 1 & 4 & -2 \\
A= \left( \begin{array}{ccc} 1 & 4 & -2 \\
to:
$$ A= \left( \begin{array}{ccc} 1 & 4 & -2 \\
Deleted line 3:
Deleted line 5:
Deleted line 6:
Added lines 1-13:
Bilde für die folgenden Matrizen alle möglichen Matrixprodukte:
$$
A= \left( \begin{array}{ccc} 1 & 4 & -2 \\
3 & 5 & 0 \end{array} \right) \; ,\;
B= \left( \begin{array}{cc} 3 & -1 \\
0 & 2 \end{array} \right) \; ,\;
C=\left( \begin{array}{ccc} 4 & 1 & 8 \\ 5 & 7 & 0 \\ 0 & 0 & 3 \end{array} \right) \; ,\;D= \left( \begin{array}{c} -2 \\ 8 \\ 1 \end{array} \right) \; ,\;
E = \left( \begin{array}{cc} 3 & 2
\end{array} \right)
$$
$$
A= \left( \begin{array}{ccc} 1 & 4 & -2 \\
3 & 5 & 0 \end{array} \right) \; ,\;
B= \left( \begin{array}{cc} 3 & -1 \\
0 & 2 \end{array} \right) \; ,\;
C=\left( \begin{array}{ccc} 4 & 1 & 8 \\ 5 & 7 & 0 \\ 0 & 0 & 3 \end{array} \right) \; ,\;D= \left( \begin{array}{c} -2 \\ 8 \\ 1 \end{array} \right) \; ,\;
E = \left( \begin{array}{cc} 3 & 2
\end{array} \right)
$$