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<!DOCTYPE html> <html lang="en"> <head> <title>genIm -- a basis of the image of a linear expression</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="differential algebra" href="index.html">DiffAlg</a> :: <a title="a basis of the image of a linear expression" href="_gen__Im.html">genIm</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> <a href="_gen__Ker.html">next</a> | <a href="_dist.html">previous</a> | <a href="_gen__Ker.html">forward</a> | <a href="_dist.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>genIm -- a basis of the image of a linear expression</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">genIm(expr,var)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">expr</span>, <span>an instance of the type <a title="the class of all differential forms and vector fields" href="___Diff__Alg__Element.html">DiffAlgElement</a></span>, an expression linear in the variable <span class="tt">var</span></span></li> <li><span><span class="tt">var</span>, <span>an instance of the type <a title="the class of all differential forms and vector fields" href="___Diff__Alg__Element.html">DiffAlgElement</a></span>, this is the variable of the linear expression, it must have free and linear scalar coefficients</span></li> </ul> </li> <li>Outputs: <ul> <li><span><span>a <a title="the class of all lists -- {...}" href="../../Macaulay2Doc/html/___List.html">list</a></span>, a basis of the image of the linear expression</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <p>This routine returns a basis of the image of <span class="tt">expr</span>, an homogeneous expression linear in <span class="tt">var</span>.</p> <br> <p>In this example we compute a basis of the image of the derivative of a projective differential 1-forms.</p> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : h = newForm(2,1,2,"a") 2 2 2 2 o1 = (a x + a x x + a x + a x x + a x x + a x )dx + (a x + a x x + 0 0 3 0 1 9 1 6 0 2 12 1 2 15 2 0 1 0 4 0 1 ------------------------------------------------------------------------ 2 2 2 2 a x + a x x + a x x + a x )dx + (a x + a x x + a x + a x x + 10 1 7 0 2 13 1 2 16 2 1 2 0 5 0 1 11 1 8 0 2 ------------------------------------------------------------------------ 2 a x x + a x )dx 14 1 2 17 2 2 o1 : DiffAlgForm</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : R = radial 2 o2 = x ax + x ax + x ax 0 0 1 1 2 2 o2 : DiffAlgField</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : H = linearComb(genKer(R _ h, h),"a") 2 2 2 o3 = (a x x + a x + a x x + a x x + a x )dx + (- a x - a x x + a x x 0 0 1 3 1 1 0 2 4 1 2 6 2 0 0 0 3 0 1 2 0 2 ------------------------------------------------------------------------ 2 2 2 + a x x + a x )dx + (- a x + (- a - a )x x - a x - a x x - 5 1 2 7 2 1 1 0 2 4 0 1 5 1 6 0 2 ------------------------------------------------------------------------ a x x )dx 7 1 2 2 o3 : DiffAlgForm</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : genIm(diff H,H) o4 = {-3x dx dx , -3x dx dx , x dx dx - x dx dx - 2x dx dx , -3x dx dx , - 0 0 1 0 0 2 2 0 1 1 0 2 0 1 2 1 0 1 ------------------------------------------------------------------------ x dx dx - 2x dx dx - x dx dx , -3x dx dx , -3x dx dx , -3x dx dx } 2 0 1 1 0 2 0 1 2 1 1 2 2 0 2 2 1 2 o4 : List</code></pre> </td> </tr> </table> <p>It is possible to get a linearly independent set of elements using this routine:</p> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i5 : w1=random newForm(2,1,2,"a");</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : w2=random newForm(2,1,2,"a");</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : w3=w1+w2;</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : u=linearComb({w1,w2,w3},"a");</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : genIm(u,u) 2 2 2 2 o9 = {(2x - 2x x + 6x - 3x x + 6x x )dx + (- 7x + 5x x - x - 2x x - 0 0 1 1 0 2 1 2 0 0 0 1 1 0 2 ----------------------------------------------------------------------- 2 2 2 2 x x + x )dx + (- 3x - 4x x + 7x + 2x x - 7x x - 3x )dx 1 2 2 1 0 0 1 1 0 2 1 2 2 2 ------------------------------------------------------------------------ 2 2 2 2 2 , (x + 7x x + 3x + 2x x + x x + 8x )dx + (- x - 4x - 7x x - x 0 0 1 1 0 2 1 2 2 0 0 1 0 2 1 - --------------------------------------------------------------------- 2 2 2 2 - x )dx + (3x - 4x x + 7x - 5x x + x )dx 2 1 0 0 1 1 0 2 2 2 ------------------------------------------------------------------------ x } 2 --- o9 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i10 : #oo o10 = 2</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="basis of the kernel of a linear expression" href="_gen__Ker.html">genKer</a> -- basis of the kernel of a linear expression</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">genIm</span>:</h2> <ul> <li><kbd>genIm(DiffAlgElement,DiffAlgElement)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="a basis of the image of a linear expression" href="_gen__Im.html">genIm</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function.html">method function</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">DiffAlg.m2:948:0</span>.</p> </div> </div> </div> </body> </html>
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