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      <header alignment="Center" color="#a9a9a9">&amp;[DATE] &amp;[TIME] - &amp;[FILENAME]</header>
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  <region id="0" left="18" top="18" width="169" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p bold="true">Lineare Gleichungen</p>
    </text>
    <text lang="eng">
      <p bold="true">Linear equations</p>
    </text>
  </region>
  <region id="1" left="594" top="18" width="94" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Mathematik.sm">Mathematik</p>
    </text>
    <text lang="eng">
      <p href="Mathematik.sm">Mathematics</p>
    </text>
  </region>
  <region id="2" left="594" top="36" width="61" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Handbook.sm">Inhalt</p>
    </text>
    <text lang="eng">
      <p href="Handbook.sm">Contents</p>
    </text>
  </region>
  <region id="3" left="18" top="54" width="98" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p underline="true">Grundlagen </p>
    </text>
    <text lang="eng">
      <p underline="true">Basics</p>
    </text>
  </region>
  <region id="4" left="18" top="90" width="686" height="56" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Lineare Gleichungen sind solche, in denen die Unbekannten nur durch Addition oder
Multiplikation mit konstanten Faktoren (Koeffizienten) miteinander verknüpft sind.
</p>
    </text>
    <text lang="eng">
      <p>Linear equations combine the unknown quantities by addition or multiplication by a 
constant (linear operations) only.</p>
    </text>
  </region>
  <region id="5" left="18" top="144" width="727" height="56" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Mehrere Gleichungen für einen gemeinsamen Satz Unbekannter nennt man Gleichungssystem.
Damit die Unbekannten eindeutig berechnet werden können, benötigt man bei n Unbekannten
auch mindestens n Gleichungen.</p>
    </text>
    <text lang="eng">
      <p>Multiple equations for a common set of unknowns are called a system of equations. In 
order to solve for n unknowns, n equations are required.</p>
    </text>
  </region>
  <region id="6" left="18" top="216" width="686" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>In allgemeiner Form lässt sich ein lineares Gleichungssystem mit n Unbekannten und
n Gleichungen schreiben:</p>
    </text>
    <text lang="eng">
      <p>Any linear system with n equations can be written in general form in this way:</p>
    </text>
  </region>
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    <math>
      <input>
        <e type="operand">i</e>
        <e type="operand">1</e>
        <e type="operand">n</e>
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  <region id="9" left="27" top="324" width="34" height="32" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">x</e>
        <e type="operand">j</e>
        <e type="function" preserve="true" args="2">el</e>
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  <region id="10" left="81" top="324" width="98" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Unbekannte </p>
    </text>
    <text lang="eng">
      <p>unknowns</p>
    </text>
  </region>
  <region id="11" left="27" top="360" width="20" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">i</e>
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    </math>
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  <region id="12" left="81" top="360" width="148" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Gleichungsnummer </p>
    </text>
    <text lang="eng">
      <p>equation counter</p>
    </text>
  </region>
  <region id="13" left="27" top="396" width="48" height="32" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">C</e>
        <e type="operand">i</e>
        <e type="operand">j</e>
        <e type="function" preserve="true" args="3">el</e>
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  <region id="14" left="81" top="396" width="480" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Bekannte Koeffizienten der Unbekannten x.j in Gleichung i</p>
    </text>
    <text lang="eng">
      <p>knwon coefficients for unknown x.j in equation i</p>
    </text>
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  <region id="15" left="27" top="432" width="34" height="32" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">r</e>
        <e type="operand">i</e>
        <e type="function" preserve="true" args="2">el</e>
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  <region id="16" left="81" top="432" width="504" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Bekannte absolute Glieder oder rechte Seiten der Gleichung i</p>
    </text>
    <text lang="eng">
      <p>known absolute term or right hand side of equation i</p>
    </text>
  </region>
  <region id="17" left="18" top="468" width="397" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Für n=2 sieht das System ausgeschrieben so aus:</p>
    </text>
    <text lang="eng">
      <p>for n=2 this can be expanded to </p>
    </text>
  </region>
  <region id="18" left="36" top="504" width="190" height="32" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">C</e>
        <e type="operand">1</e>
        <e type="operand">1</e>
        <e type="function" preserve="true" args="3">el</e>
        <e type="operand">x</e>
        <e type="operand">1</e>
        <e type="function" preserve="true" args="2">el</e>
        <e type="operator" args="2">*</e>
        <e type="operand">C</e>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="3">el</e>
        <e type="operand">x</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="2">el</e>
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  <region id="19" left="36" top="531" width="190" height="32" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">C</e>
        <e type="operand">2</e>
        <e type="operand">1</e>
        <e type="function" preserve="true" args="3">el</e>
        <e type="operand">x</e>
        <e type="operand">1</e>
        <e type="function" preserve="true" args="2">el</e>
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        <e type="operand">C</e>
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        <e type="operand">2</e>
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  <region id="20" left="18" top="576" width="641" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Mit den Regeln für die Matrixmultiplikation kann man das Gleichungssystem in 
Matrixform schreiben:</p>
    </text>
    <text lang="eng">
      <p>According to the rules of matrix multiplication the system of equations can be 
written in matrix format</p>
    </text>
  </region>
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    <math>
      <input>
        <e type="operand">C</e>
        <e type="operand">1</e>
        <e type="operand">1</e>
        <e type="function" preserve="true" args="3">el</e>
        <e type="operand">C</e>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="3">el</e>
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        <e type="operand">1</e>
        <e type="function" preserve="true" args="3">el</e>
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        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="3">el</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
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        <e type="operand">1</e>
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        <e type="operand">1</e>
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        <e type="function" preserve="true" args="2">el</e>
        <e type="operand">r</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="2">el</e>
        <e type="operand">2</e>
        <e type="operand">1</e>
        <e type="function" preserve="true" args="4">mat</e>
        <e type="operator" args="2">≡</e>
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  </region>
  <region id="22" left="234" top="648" width="300" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>oder mit Symbolen für die Matrizen:</p>
    </text>
    <text lang="eng">
      <p>or more compact with symbols for 
the matrices</p>
    </text>
  </region>
  <region id="23" left="549" top="648" width="56" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">C</e>
        <e type="operand">x</e>
        <e type="operator" args="2">*</e>
        <e type="operand">r</e>
        <e type="operator" args="2">≡</e>
      </input>
    </math>
  </region>
  <region id="24" left="18" top="702" width="672" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Die Lösung erhält man, wenn man diese Gleichung mit der Inversen von C von links 
multipliziert:</p>
    </text>
    <text lang="eng">
      <p>The solution is obtained by left side multiplying the equation by the inverse of C</p>
    </text>
  </region>
  <region id="25" left="36" top="747" width="128" height="33" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">C</e>
        <e type="operand">1</e>
        <e type="operator" args="1">-</e>
        <e type="operator" args="2">^</e>
        <e type="operand">C</e>
        <e type="operator" args="2">*</e>
        <e type="operand">x</e>
        <e type="operator" args="2">*</e>
        <e type="operand">C</e>
        <e type="operand">1</e>
        <e type="operator" args="1">-</e>
        <e type="operator" args="2">^</e>
        <e type="operand">r</e>
        <e type="operator" args="2">*</e>
        <e type="operator" args="2">≡</e>
      </input>
    </math>
  </region>
  <region id="26" left="243" top="747" width="77" height="33" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">x</e>
        <e type="operand">C</e>
        <e type="operand">1</e>
        <e type="operator" args="1">-</e>
        <e type="operator" args="2">^</e>
        <e type="operand">r</e>
        <e type="operator" args="2">*</e>
        <e type="operator" args="2">≡</e>
      </input>
    </math>
  </region>
  <region id="27" left="198" top="756" width="32" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>-&gt; </p>
    </text>
  </region>
  <region id="28" left="18" top="792" width="673" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Die Voraussetzung dafür ist, dass die Determinante von C nicht verschwindet (das 
wäre analog zur Division durch Null).</p>
    </text>
    <text lang="eng">
      <p>This requires the determinant of C to be nonzero, otherwise this would be similar to
division by zero.</p>
    </text>
  </region>
  <region id="29" left="18" top="846" width="217" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p underline="true">Lösungsverfahren in SMath</p>
    </text>
    <text lang="eng">
      <p underline="true">Solution methods in SMath</p>
    </text>
  </region>
  <region id="30" left="18" top="873" width="647" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Grundsätzlich haben Sie in SMath Studio drei verschiedene Optionen zur Lösung 
linearer Gleichungssysteme:</p>
    </text>
    <text lang="eng">
      <p>There are basically three options for solving linear systems in SMath:</p>
    </text>
  </region>
  <region id="31" left="18" top="927" width="702" height="56" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>1. Multiplikation der inversen Koeffizientenmatrix mit der rechten Seite. Dafür muss
      das Gleichungssystem in Matrixform (Matrix C und Vektor r) vorliegen. Mit dieser
      Variante können Sie auch symbolisch rechnen.</p>
    </text>
    <text lang="eng">
      <p>1.  Multiplication of the right hand side by the inverse coefficient matrix. This 
       requires the system to be available as matrix C and vector r. This option can 
       provide symbiolic solutions  </p>
    </text>
  </region>
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    <math>
      <input>
        <e type="operand">a</e>
        <e type="operand">b</e>
        <e type="operand">c</e>
        <e type="operand">d</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
        <e type="operand">1</e>
        <e type="operator" args="1">-</e>
        <e type="operator" args="2">^</e>
        <e type="operand">e</e>
        <e type="operand">f</e>
        <e type="operand">2</e>
        <e type="operand">1</e>
        <e type="function" preserve="true" args="4">mat</e>
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        <e type="operand">d</e>
        <e type="operand">e</e>
        <e type="operator" args="2">*</e>
        <e type="operator" args="1">-</e>
        <e type="operand">b</e>
        <e type="operand">f</e>
        <e type="operator" args="2">*</e>
        <e type="operator" args="2">+</e>
        <e type="operand">b</e>
        <e type="operand">c</e>
        <e type="operator" args="2">*</e>
        <e type="operand">a</e>
        <e type="operand">d</e>
        <e type="operator" args="2">*</e>
        <e type="operator" args="2">-</e>
        <e type="operator" args="2">/</e>
        <e type="operand">e</e>
        <e type="operand">c</e>
        <e type="operator" args="2">*</e>
        <e type="operand">a</e>
        <e type="operand">f</e>
        <e type="operator" args="2">*</e>
        <e type="operator" args="2">-</e>
        <e type="operand">b</e>
        <e type="operand">c</e>
        <e type="operator" args="2">*</e>
        <e type="operand">a</e>
        <e type="operand">d</e>
        <e type="operator" args="2">*</e>
        <e type="operator" args="2">-</e>
        <e type="operator" args="2">/</e>
        <e type="operand">2</e>
        <e type="operand">1</e>
        <e type="function" preserve="true" args="4">mat</e>
      </result>
    </math>
  </region>
  <region id="33" left="423" top="981" width="284" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Section matrix operations.sm">Rechnen mit Matrizen und Vektoren</p>
    </text>
    <text lang="eng">
      <p href="Section matrix operations.sm">Matrix operations</p>
    </text>
  </region>
  <region id="34" left="18" top="1062" width="699" height="88" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>2. Benutzung der Techniken für nichtlineare Gleichungssysteme, insbesondere der 
      Funktion FindRoot(). Das Gleichungssystem muss als Vektor von Gleichungen oder
      Null zu setzenden Ausdrücken vorliegen. Die Variablen, nach denen aufgelöst wird,
      benötigen einen Startwert. Die Funktion kann die Variablen direkt mit der Lösung
      belegen (eingebettete Zuweisung).  </p>
    </text>
    <text lang="eng">
      <p>2. Use of the techniques for nonlinear systems of equations, in particular the 
      function FindRoot(). The system must be provided as vector of equations or 
      expressions to find the roots of. The unknowns to be solved for need initial 
      values of correct physical dimension. The function can assign the solution directly
      to the variables (embedded assignment)</p>
    </text>
  </region>
  <region id="35" left="423" top="1134" width="94" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="findroot.sm">FindRoot()</p>
    </text>
  </region>
  <region id="36" left="522" top="1134" width="210" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Section NLsolve.sm">Nichtlineare Gleichungen</p>
    </text>
    <text lang="eng">
      <p href="Section NLsolve.sm">Solving Non-Linear Equations</p>
    </text>
  </region>
  <region id="37" left="18" top="1179" width="684" height="72" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>3. Funktion LinSolve() aus dem Maxima-Plugin Das Gleichungssystem wird 
      genau wie bei FindRoot() als Vektor von Gleichungen dargestellt. Startwerte
      werden nicht benötigt. Das Ergebnis wird in Form von Gleichungen zurückgegeben;
      Diese Gleichungen können mit Assign() für die Zuweisung benutzt werden. </p>
    </text>
    <text lang="eng">
      <p>3. Function LinSolve() from the LinSolve snippet. The system is specified the same
      way as with FindRoot() as a vector of equations. No initial values are required
      but embedded assignment is not available.</p>
    </text>
  </region>
  <region id="38" left="36" top="1260" width="65" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Option </p>
    </text>
  </region>
  <region id="39" left="189" top="1260" width="20" height="264" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>│
│
│
│
│
│
│
│
│
│
│
│
│
│
│
</p>
    </text>
  </region>
  <region id="40" left="207" top="1260" width="61" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Matrix</p>
    </text>
  </region>
  <region id="41" left="333" top="1260" width="20" height="264" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>│
│
│
│
│
│
│
│
│
│
│
│
│
│
│
</p>
    </text>
  </region>
  <region id="42" left="351" top="1260" width="94" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>FindRoot()</p>
    </text>
  </region>
  <region id="43" left="477" top="1260" width="20" height="264" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>│
│
│
│
│
│
│
│
│
│
│
│
│
│
│
</p>
    </text>
  </region>
  <region id="44" left="495" top="1260" width="94" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>LinSolve()</p>
    </text>
  </region>
  <region id="45" left="18" top="1269" width="597" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>_______________________________________________________________________</p>
    </text>
  </region>
  <region id="46" left="36" top="1296" width="127" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>System gegeben
als:</p>
    </text>
    <text lang="eng">
      <p>System given as</p>
    </text>
  </region>
  <region id="47" left="207" top="1296" width="127" height="72" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Koeffizienten-
matrix,
Vektor der 
rechten Seite</p>
    </text>
    <text lang="eng">
      <p>coefficient 
matrix, 
RHS vector</p>
    </text>
  </region>
  <region id="48" left="351" top="1296" width="103" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Vektor mit 
Gleichungen</p>
    </text>
    <text lang="eng">
      <p>vector of 
boolean 
equations</p>
    </text>
  </region>
  <region id="49" left="495" top="1296" width="103" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Vektor mit 
Gleichungen</p>
    </text>
    <text lang="eng">
      <p>vector of 
boolean 
equations</p>
    </text>
  </region>
  <region id="50" left="36" top="1377" width="115" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Startwerte
erforderlich </p>
    </text>
    <text lang="eng">
      <p>Initial guesses
required</p>
    </text>
  </region>
  <region id="51" left="207" top="1377" width="49" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Nein </p>
    </text>
    <text lang="eng">
      <p>No</p>
    </text>
  </region>
  <region id="52" left="351" top="1377" width="32" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Ja </p>
    </text>
    <text lang="eng">
      <p>Yes</p>
    </text>
  </region>
  <region id="53" left="495" top="1377" width="49" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Nein </p>
    </text>
    <text lang="eng">
      <p>No</p>
    </text>
  </region>
  <region id="54" left="36" top="1422" width="152" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>eingebettete 
Ergebniszuweisung</p>
    </text>
    <text lang="eng">
      <p>embedded results
assignment</p>
    </text>
  </region>
  <region id="55" left="207" top="1422" width="49" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Nein </p>
    </text>
    <text lang="eng">
      <p>No</p>
    </text>
  </region>
  <region id="56" left="351" top="1422" width="32" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Ja </p>
    </text>
    <text lang="eng">
      <p>Yes</p>
    </text>
  </region>
  <region id="57" left="495" top="1422" width="111" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>mit Assign()</p>
    </text>
    <text lang="eng">
      <p>No</p>
    </text>
  </region>
  <region id="58" left="36" top="1467" width="107" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Symbolische 
Lösung</p>
    </text>
  </region>
  <region id="59" left="207" top="1467" width="28" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">Ja</e>
      </input>
    </math>
  </region>
  <region id="60" left="351" top="1467" width="119" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>indirekt, mit
Maßeinheiten</p>
    </text>
  </region>
  <region id="61" left="495" top="1467" width="28" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">Ja</e>
      </input>
    </math>
  </region>
  <region id="62" left="18" top="1530" width="505" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="example support reactions linear system.sm">Beispiel: Auflagerreaktionen berechnen (lineare Gleichungen)</p>
    </text>
    <text lang="eng">
      <p href="example support reactions linear system.sm">Example: Compute support reactions (linear system)</p>
    </text>
  </region>
  <region id="63" left="18" top="1584" width="94" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Mathematik.sm">Mathematik</p>
    </text>
    <text lang="eng">
      <p href="Mathematik.sm">Mathematics</p>
    </text>
  </region>
  <region id="64" left="18" top="1602" width="61" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Handbook.sm">Inhalt</p>
    </text>
    <text lang="eng">
      <p href="Handbook.sm">Contents</p>
    </text>
  </region>
  <region id="65" left="18" top="1638" width="450" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p italic="true">$Author: mkraska $
$Date: 2013-10-08 23:44:40 +0200 (Di, 08. Okt 2013) $ </p>
    </text>
  </region>
</regions>