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<head><title>matrices to and from modules -- including kernel, cokernel and image</title>
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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_modules.html" title="">modules</a> > <a href="_matrices_spto_spand_spfrom_spmodules.html" title="including kernel, cokernel and image">matrices to and from modules</a></div>
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<div><h1>matrices to and from modules -- including kernel, cokernel and image</h1>
<div><h2>matrices to modules (kernel, image, cokernel)</h2>
Given a matrix, we may compute the <a href="_kernel.html" title="kernel of a ringmap, matrix, or chain complex">kernel</a>, <a href="_image.html" title="image of a map">image</a>, and <a href="_cokernel.html" title="cokernel of a map of modules, graded modules, or chaincomplexes">cokernel</a>.<table class="examples"><tr><td><pre>i1 : R = QQ[a..f];</pre>
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<tr><td><pre>i2 : F = matrix{{a,b,d,e},{b,c,e,f}}
o2 = | a b d e |
| b c e f |
2 4
o2 : Matrix R <--- R</pre>
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<table class="examples"><tr><td><pre>i3 : M = ker F
o3 = image {1} | cd-be 0 e2-df ce-bf |
{1} | -bd+ae e2-df 0 -be+af |
{1} | b2-ac -ce+bf -be+af 0 |
{1} | 0 cd-be bd-ae b2-ac |
4
o3 : R-module, submodule of R</pre>
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<tr><td><pre>i4 : coker F
o4 = cokernel | a b d e |
| b c e f |
2
o4 : R-module, quotient of R</pre>
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<tr><td><pre>i5 : image F
o5 = image | a b d e |
| b c e f |
2
o5 : R-module, submodule of R</pre>
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Some routines in Macaulay2 have abbreviations, for example <tt>ker</tt> may be used for <tt>kernel</tt>, and <tt>coker</tt> may be used for <tt>cokernel</tt>. The <tt>image</tt> function has no abbreviated form.<h2>modules to matrices</h2>
Each module has, at least implicitly, two matrices associated to it: <a href="_generators.html" title="provide matrix or list of generators">generators</a> (abbreviated form: <tt>gens</tt>), and <a href="_relations.html" title="the defining relations">relations</a>. If a module is a submodule of a free module, then the relations matrix is zero. If a module is a quotient of a free module, then the generator matrix is the identity matrix. If a module is a <a href="_subquotient.html" title="make a subquotient module">subquotient</a>, then both may be more general.<table class="examples"><tr><td><pre>i6 : generators M
o6 = {1} | cd-be 0 e2-df ce-bf |
{1} | -bd+ae e2-df 0 -be+af |
{1} | b2-ac -ce+bf -be+af 0 |
{1} | 0 cd-be bd-ae b2-ac |
4 4
o6 : Matrix R <--- R</pre>
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<tr><td><pre>i7 : relations M
o7 = 0
4
o7 : Matrix R <--- 0</pre>
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Every finitely generated module has a presentation matrix. In Macaulay2, if the module is not a quotient of a free module, then a syzygy computation is performed to find a presentation matrix.<table class="examples"><tr><td><pre>i8 : presentation M
o8 = {3} | -f -e |
{3} | b a |
{3} | -c -b |
{3} | e d |
4 2
o8 : Matrix R <--- R</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_cokernel.html" title="cokernel of a map of modules, graded modules, or chaincomplexes">cokernel(Matrix)</a> -- cokernel of a map of modules, graded modules, or chaincomplexes</span></li>
<li><span><a href="_image.html" title="image of a map">image(Matrix)</a> -- image of a map</span></li>
<li><span><a href="_kernel_lp__Matrix_rp.html" title="kernel of a matrix">kernel(Matrix)</a> -- kernel of a matrix</span></li>
<li><span><a href="_generators_lp__Module_rp.html" title="the generator matrix of a module">generators(Module)</a> -- the generator matrix of a module</span></li>
<li><span><a href="_relations.html" title="the defining relations">relations(Module)</a> -- the defining relations</span></li>
<li><span><a href="_presentation_lp__Module_rp.html" title="presentation of a module">presentation(Module)</a> -- presentation of a module</span></li>
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