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<head><title>isInjective -- whether a map is injective</title>
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<div><h1>isInjective -- whether a map is injective</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>isInjective f</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>f</tt>, <span>a <a href="___Matrix.html">matrix</a></span>, or <span>a <a href="___Ring__Map.html">ring map</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Boolean.html">Boolean value</a></span>, whether the kernel is zero</span></li>
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<div class="single"><h2>Description</h2>
<div>This function computes the kernel, and checks whether it is zero.<table class="examples"><tr><td><pre>i1 : R = QQ[a..d];</pre>
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<tr><td><pre>i2 : F = matrix{{a,b},{c,d}}
o2 = | a b |
| c d |
2 2
o2 : Matrix R <--- R</pre>
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<tr><td><pre>i3 : isInjective F
o3 = true</pre>
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<tr><td><pre>i4 : G = substitute(F, R/(det F))
o4 = | a b |
| c d |
R 2 R 2
o4 : Matrix (-----------) <--- (-----------)
- b*c + a*d - b*c + a*d</pre>
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<tr><td><pre>i5 : isInjective G
o5 = false</pre>
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Similarly for ring maps:<table class="examples"><tr><td><pre>i6 : S = QQ[r,s,t];</pre>
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<tr><td><pre>i7 : phi = map(S,R,{r^3, r^2*s, r*s*t, s^3})
3 2 3
o7 = map(S,R,{r , r s, r*s*t, s })
o7 : RingMap S <--- R</pre>
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<tr><td><pre>i8 : isInjective phi
o8 = false</pre>
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<tr><td><pre>i9 : S' = coimage phi
o9 = S'
o9 : QuotientRing</pre>
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<tr><td><pre>i10 : phi' = phi * map(R,S')
3 2 3
o10 = map(S,S',{r , r s, r*s*t, s })
o10 : RingMap S <--- S'</pre>
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<tr><td><pre>i11 : isInjective phi'
o11 = true</pre>
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<div class="single"><h2>Caveat</h2>
<div>One could imagine a faster routine for this. If you write one, please send it to us!</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_kernel.html" title="kernel of a ringmap, matrix, or chain complex">kernel</a> -- kernel of a ringmap, matrix, or chain complex</span></li>
<li><span><a href="_is__Surjective.html" title="whether a map is surjective">isSurjective</a> -- whether a map is surjective</span></li>
<li><span><a href="_coimage.html" title="coimage of a map">coimage(RingMap)</a> -- coimage of a map</span></li>
<li><span><a href="_determinant.html" title="determinant of a matrix">determinant</a> -- determinant of a matrix</span></li>
<li><span><a href="_substitution_spand_spmaps_spbetween_springs.html" title="">substitution and maps between rings</a></span></li>
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<div class="waystouse"><h2>Ways to use <tt>isInjective</tt> :</h2>
<ul><li>isInjective(Matrix)</li>
<li>isInjective(RingMap)</li>
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