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<p>&nbsp;<p>
<hr noshade size="5">
<h2><a name="MatKernel"></a>
MatKernel
</h2>

Returns a basis for the kernel of the given matrix.

<p><p>

<h3>
Syntax:
</h3>
<pre>K := MatKernel(M);
K := MatKernel(M, d);</pre>

<p>

<table>
<tr>
<td>matrix</td>
<td><pre>  K  </pre></td>
<td></td>
</tr>
<tr>
<td>matrix</td>
<td><pre>  M  </pre></td>
<td></td>
</tr>
<tr>
<td>integer</td>
<td><pre>  d  </pre></td>
<td></td>
</tr>
</table>

<p>


<h3>
Description:
</h3>
Let <tt> M</tt> be a m \times n-matrix. The MatKernel routine
computes a basis <tt> K</tt> for the linear subspace
 { x = (x|1, &hellip; ,x|m) : x cdot <tt> M</tt> = (0, &hellip; ,0) }.
.

If a second parameter d is used, the kernel
over Z / dZ will be computed. The used algorithm is based
on BuNei1.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
Compute the kernel of the integer matrix
\left(\begin{array}{ccc} 2 & 3 & 4 3 & 4 & 5
4 & 5 & 6 6 & 7 & 8 \end{array}right). 
<pre>

kash> M := Mat(Z,[[2,3,4],[3,4,5],[4,5,6],[6,7,8]]);
[2 3 4]
[3 4 5]
[4 5 6]
[6 7 8]
kash> MatKernel(M);
[ 1  0 -2  1]
[ 0  2 -3  1]


</pre>

<p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
 Compute the kernel modulo 3 
<pre>

kash> MatKernel(M, 3);
[0 2 0 1]
[1 1 1 0]


</pre>

<p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
 Compute kernel
<pre>

kash> o := Order(Z, 2,3);;
kash> a := Elt(o, [1,2/19]);;
kash> one := Elt(o, [1,0]);;
kash> m := Mat(o, [[a,2*a,3*a^2],[a^2,2*a^2,3*a^3],[one,2*one,3*a]]);
[[19, 2] / 19 [38, 4] / 19 [1119, 228] / 361]
[[373, 76] / 361 [746, 152] / 361 [22629, 6570] / 6859]
[1 2 [57, 6] / 19]
kash> k := MatKernel(m);
[1 0 [-19, -2] / 19]
[0 1 [-373, -76] / 361]
kash> k*m;
> [0 0 0]
[0 0 0]

</pre>

<p>


<hr>

<p>
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