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<head><title>kernel(Matrix) -- kernel of a matrix</title>
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<div><h1>kernel(Matrix) -- kernel of a matrix</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>kernel f</tt></div>
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<li><span>Function: <a href="_kernel.html" title="kernel of a ringmap, matrix, or chain complex">kernel</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>f</tt>, <span>a <a href="___Matrix.html">matrix</a></span>, a map of modules <tt>M --> N</tt></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Module.html">module</a></span>, the kernel of f, a submodule of M</span></li>
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<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_kernel_lp..._cm_sp__Subring__Limit_sp_eq_gt_sp..._rp.html">SubringLimit => ...</a>, </span></li>
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<div class="single"><h2>Description</h2>
<div>If f is <span>a <a href="___Ring__Element.html">ring element</a></span>, then it will be interpreted as a one by one matrix.<p/>
The kernel is the submodule of M of all elements mapping to zero under <tt>f</tt>. Over polynomial rings, this is computed using a Gröbner basis computation.<table class="examples"><tr><td><pre>i1 : R = ZZ/32003[vars(0..10)]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : M = genericSkewMatrix(R,a,5)
o2 = | 0 a b c d |
| -a 0 e f g |
| -b -e 0 h i |
| -c -f -h 0 j |
| -d -g -i -j 0 |
5 5
o2 : Matrix R <--- R</pre>
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<tr><td><pre>i3 : ker M
o3 = image {1} | gh-fi+ej |
{1} | -dh+ci-bj |
{1} | df-cg+aj |
{1} | -de+bg-ai |
{1} | ce-bf+ah |
5
o3 : R-module, submodule of R</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_syz.html" title="the syzygy matrix">syz</a> -- the syzygy matrix</span></li>
<li><span><a href="_generic__Skew__Matrix.html" title="make a generic skew symmetric matrix of variables">genericSkewMatrix</a> -- make a generic skew symmetric matrix of variables</span></li>
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