/usr/share/doc/axiom-doc/hypertex/equsystemlinear.xhtml

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  <div align="center"><img align="middle" src="doctitle.png"/></div>
  <hr/>
  <div align="center">Solution of Systems of Linear Equations</div>
  <hr/>
You can use the operation <a href="dbopsolve.xhtml">solve</a> to solve
systems of linear equations.

The operation <a href="dbopsolve.xhtml">solve</a> takes two arguments, the
list of equations and the list of the unknowns to be solved for. A system
of linear equations need not have a unique solution.

To solve the linear system:
<pre>
        x + y + x = 8
    3*x - 2*y + z = 0
    x + 2*y + 2*z = 17
</pre>
evaluate this expression.
<ul>
 <li>
  <input type="submit" id="p1" class="subbut" 
    onclick="makeRequest('p1');"
    value="solve([x+y+x=8,3*x-2*y+z=0,x+2*y+2*z=17],[x,y,z])" />
  <div id="ansp1"><div></div></div>
 </li>
</ul>
Parameters are given as new variables starting with a percent sign and
"%" and the variables are expressed in terms of the parameters. If the system
has no solutions then the empty list is returned.

When you solve the linear system
<pre>
      x + 2*y + 3*z = 2
    2*x + 3*y + 4*z = 2
    3*x + 4*y + 5*z = 2
</pre>
with this expression you get a solution involving a parameter.
<ul>
 <li>
  <input type="submit" id="p2" class="subbut" 
    onclick="makeRequest('p2');"
    value="solve([x+2*y+3*z=2,2*x+3*y+4*z=2,3*x+4*y+5*z=2],[x,y,z])" />
  <div id="ansp2"><div></div></div>
 </li>
</ul>
The system can also be presented as a matrix and a vector. The matrix 
contains the coefficients of the linear equations and the vector contains
the numbers appearing on the right-hand sides of the equations. You may 
input the matrix as a list of rows and the vector as a list of its elements.

To solve the system:
<pre>
       x + y + z = 8
   2*x - 2*y + z = 0
   x + 2*y + 2*z = 17
</pre>
in matrix form you would evaluate this expression.
<ul>
 <li>
  <input type="submit" id="p3" class="subbut" 
    onclick="makeRequest('p3');"
    value="solve([[1,1,1],[3,-2,1],[1,2,2]],[8,0,17])" />
  <div id="ansp3"><div></div></div>
 </li>
</ul>
The solutions are presented as a Record with two components: the component
particular contains a particular solution of the given system or the item
"failed" if there are no solutions, the component basis contains a list of
vectors that are a basis for the space of solutions of the corresponding
homogeneous system. If the system of linear equations does not have a unique
solution, then the basis component contains non-trivial vectors.

This happens when you solve the linear system
<pre>
    x + 2*y + 3*z = 2
  2*x + 3*y + 4*z = 2
  3*x + 4*y + 5*z = 2
</pre>
with this command.
<ul>
 <li>
  <input type="submit" id="p4" class="subbut" 
    onclick="makeRequest('p4');"
    value="solve([[1,2,3],[2,3,4],[3,4,5]],[2,2,2])" />
  <div id="ansp4"><div></div></div>
 </li>
</ul>
All solutions of this system are obtained by adding the particular solution
with a linear combination of the basis vectors.

When no solution exists then "failed" is returned as the particular 
component, as follows:
<ul>
 <li>
  <input type="submit" id="p5" class="subbut" 
    onclick="makeRequest('p5');"
    value="solve([[1,2,3],[2,3,4],[3,4,5]],[2,3,2])" />
  <div id="ansp5"><div></div></div>
 </li>
</ul>
When you want to solve a system of homogeneous equations (that is, a system
where the numbers on the right-hand sides of the equations are all zero)
in the matrix form you can omit the second argument and use the 
<a href="dbopnullspace.xhtml">nullSpace</a> operation.

This computes the solutions of the following system of equations:
<pre>
    x + 2*y + 3*z = 0
  2*x + 3*y + 4*z = 0
  3*x + 4*y + 5*z = 0
</pre>
The result is given as a list of vectors and these vectors form a basis for
the solution space.
<ul>
 <li>
  <input type="submit" id="p6" class="subbut" 
    onclick="makeRequest('p6');"
    value="nullSpace([[1,2,3],[2,3,4],[3,4,5]])" />
  <div id="ansp6"><div></div></div>
 </li>
</ul>

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</html>
axiom-hypertex-data 20120501-8 / usr / share / doc / axiom-doc / hypertex / equsystemlinear.xhtml
