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		<div id="siteNotice"><script type='text/javascript'>if (wgNotice != '') document.writeln(wgNotice);</script></div>		<h1 class="firstHeading">Bilinear map</h1>
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			<h3 id="siteSub">From Wikipedia, the free encyclopedia</h3>
			<div id="contentSub">&nbsp;&nbsp;(Redirected from <a href="http://buttnet.org/w/index.php?title=Bilinear_operator&amp;redirect=no" title="Bilinear operator">Bilinear operator</a>)</div>
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			<p>In <a href="http://buttnet.org/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>bilinear map</b> is a <a href="http://buttnet.org/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> of two arguments that is <a href="http://buttnet.org/wiki/Linear_map" title="Linear map">linear</a> in each. An example of such a map is <a href="http://buttnet.org/wiki/Multiplication" title="Multiplication">multiplication</a> of <a href="http://buttnet.org/wiki/Integers" title="Integers" class="mw-redirect">integers</a>.</p>
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<h2>Contents</h2>
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<ul>
<li class="toclevel-1"><a href="Bilinear_operator.html#Definition"><span class="tocnumber">1</span> <span class="toctext">Definition</span></a></li>
<li class="toclevel-1"><a href="Bilinear_operator.html#Properties"><span class="tocnumber">2</span> <span class="toctext">Properties</span></a></li>
<li class="toclevel-1"><a href="Bilinear_operator.html#Examples"><span class="tocnumber">3</span> <span class="toctext">Examples</span></a></li>
<li class="toclevel-1"><a href="Bilinear_operator.html#See_also"><span class="tocnumber">4</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1"><a href="Bilinear_operator.html#External_links"><span class="tocnumber">5</span> <span class="toctext">External links</span></a></li>
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<p><a name="Definition" id="Definition"></a></p>
<h2><span class="editsection">[<a href="http://buttnet.org/w/index.php?title=Bilinear_map&amp;action=edit&amp;section=1" title="Edit section: Definition">edit</a>]</span> <span class="mw-headline">Definition</span></h2>
<p>Let <i>V</i>, <i>W</i> and <i>X</i> be three <a href="http://buttnet.org/wiki/Vector_space" title="Vector space">vector spaces</a> over the same base <a href="http://buttnet.org/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> <i>F</i>. A bilinear map is a <a href="http://buttnet.org/wiki/Function_(mathematics)" title="Function (mathematics)">function</a></p>
<dl>
<dd><i>B</i>&#160;: <i>V</i> × <i>W</i> → <i>X</i></dd>
</dl>
<p>such that for any <i>w</i> in <i>W</i> the map</p>
<dl>
<dd><i>v</i> → <i>B</i>(<i>v</i>, <i>w</i>)</dd>
</dl>
<p>is a <a href="http://buttnet.org/wiki/Linear_map" title="Linear map">linear map</a> from <i>V</i> to <i>X</i>, and for any <i>v</i> in <i>V</i> the map</p>
<dl>
<dd><i>w</i> → <i>B</i>(<i>v</i>, <i>w</i>)</dd>
</dl>
<p>is a linear map from <i>W</i> to <i>X</i>.</p>
<p>In other words, if we hold the first entry of the bilinear map fixed, while letting the second entry vary, the result is a linear operator, and similarly if we hold the second entry fixed.</p>
<p>If <i>V</i> = <i>W</i> and we have <i>B</i>(<i>v</i>,<i>w</i>) = <i>B</i>(<i>w</i>,<i>v</i>) for all <i>v</i>,<i>w</i> in <i>V</i>, then we say that <i>B</i> is <i><a href="http://buttnet.org/wiki/Symmetric_function" title="Symmetric function">symmetric</a></i>.</p>
<p>The case where <i>X</i> is <i>F</i>, and we have a <b><a href="http://buttnet.org/wiki/Bilinear_form" title="Bilinear form">bilinear form</a></b>, is particularly useful (see for example <a href="http://buttnet.org/wiki/Scalar_product" title="Scalar product" class="mw-redirect">scalar product</a>, <a href="http://buttnet.org/wiki/Inner_product" title="Inner product" class="mw-redirect">inner product</a> and <a href="http://buttnet.org/wiki/Quadratic_form" title="Quadratic form">quadratic form</a>).</p>
<p>The definition works without any changes if instead of vector spaces we use <a href="http://buttnet.org/wiki/Module_(mathematics)" title="Module (mathematics)">modules</a> over a <a href="http://buttnet.org/wiki/Commutative_ring" title="Commutative ring">commutative ring</a> <i>R</i>. It also can be easily generalized to <i>n</i>-ary functions, where the proper term is <i>multilinear</i>.</p>
<p>For the case of a non-commutative base ring <i>R</i> and a right module <i>M<sub>R</sub></i> and a left module <i><sub>R</sub>N</i>, we can define a bilinear map <i>B</i>&#160;: <i>M</i> × <i>N</i> → <i>T</i>, where <i>T</i> is an abelian <a href="http://buttnet.org/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>, such that for any <i>n</i> in <i>N</i>, <i>m</i> → <i>B</i>(<i>m</i>, <i>n</i>) is a group homomorphism, and for any <i>m</i> in <i>M</i>, <i>n</i> → <i>B</i>(<i>m</i>, <i>n</i>) is a group homomorphism too, and which also satisfies</p>
<dl>
<dd><i>B</i>(<i>mt</i>, <i>n</i>) = <i>B</i>(<i>m</i>, <i>tn</i>)</dd>
</dl>
<p>for all <i>m</i> in <i>M</i>, <i>n</i> in <i>N</i> and <i>t</i> in <i>R</i>.</p>
<p><a name="Properties" id="Properties"></a></p>
<h2><span class="editsection">[<a href="http://buttnet.org/w/index.php?title=Bilinear_map&amp;action=edit&amp;section=2" title="Edit section: Properties">edit</a>]</span> <span class="mw-headline">Properties</span></h2>
<p>A first immediate consequence of the definition is that <span class="texhtml"><i>B</i>(<i>x</i>,<i>y</i>) = <i>o</i></span> whenever <i>x</i>=o or <i>y</i>=o. (This is seen by writing the <a href="http://buttnet.org/wiki/Null_vector" title="Null vector">null vector</a> <i>o</i> as 0·<i>o</i> and moving the scalar 0 "outside", in front of <i>B</i>, by linearity.)</p>
<p>The set <i>L(V,W;X)</i> of all bilinear maps is a <a href="http://buttnet.org/wiki/Linear_subspace" title="Linear subspace">linear subspace</a> of the space (<a href="http://buttnet.org/wiki/Viz." title="Viz.">viz.</a> <a href="http://buttnet.org/wiki/Vector_space" title="Vector space">vector space</a>, <a href="http://buttnet.org/wiki/Module_(mathematics)" title="Module (mathematics)">module</a>) of all maps from <i>V</i>×<i>W</i> into <i>X</i>.</p>
<p>If <i>V</i>,<i>W</i>,<i>X</i> are <a href="http://buttnet.org/wiki/Finite-dimensional" title="Finite-dimensional" class="mw-redirect">finite-dimensional</a>, then so is <i>L(V,W;X)</i>. For <i>X=F</i>, i.e. bilinear forms, the dimension of this space is dim<i>V</i>×dim<i>W</i> (while the space <i>L(V×W;K)</i> of <i>linear</i> forms is of dimension dim<i>V</i>+dim<i>W</i>). To see this, choose a <a href="http://buttnet.org/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> for <i>V</i> and <i>W</i>; then each bilinear map can be uniquely represented by the matrix <span class="texhtml"><i>B</i>(<i>e</i><sub><i>i</i></sub>,<i>f</i><sub><i>j</i></sub>)</span>, and vice versa. Now, if <i>X</i> is a space of higher dimension, we obviously have dim<i>L(V,W;X)</i>=dim<i>V</i>×dim<i>W</i>×dim<i>X</i>.</p>
<p><a name="Examples" id="Examples"></a></p>
<h2><span class="editsection">[<a href="http://buttnet.org/w/index.php?title=Bilinear_map&amp;action=edit&amp;section=3" title="Edit section: Examples">edit</a>]</span> <span class="mw-headline">Examples</span></h2>
<ul>
<li><a href="http://buttnet.org/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrix multiplication</a> is a bilinear map M(<i>m</i>,<i>n</i>) × M(<i>n</i>,<i>p</i>) → M(<i>m</i>,<i>p</i>).</li>
<li>If a <a href="http://buttnet.org/wiki/Vector_space" title="Vector space">vector space</a> <i>V</i> over the <a href="http://buttnet.org/wiki/Real_number" title="Real number">real numbers</a> <b>R</b> carries an <a href="http://buttnet.org/wiki/Inner_product_space" title="Inner product space">inner product</a>, then the inner product is a bilinear map <i>V</i> × <i>V</i> → <b>R</b>.</li>
<li>In general, for a vector space <i>V</i> over a field <i>F</i>, a <a href="http://buttnet.org/wiki/Bilinear_form" title="Bilinear form">bilinear form</a> on <i>V</i> is the same as a bilinear map <i>V</i> × <i>V</i> → <i>F</i>.</li>
<li>If <i>V</i> is a vector space with <a href="http://buttnet.org/wiki/Dual_space" title="Dual space">dual space</a> <i>V*</i>, then the application operator, <i>b</i>(<i>f</i>, <i>v</i>) = <i>f</i>(<i>v</i>) is a bilinear map from <i>V</i>* × <i>V</i> to the base field.</li>
<li>Let <i>V</i> and <i>W</i> be vector spaces over the same base field <i>F</i>. If <i>f</i> is a member of <i>V</i>* and <i>g</i> a member of <i>W</i>*, then <i>b</i>(<i>v</i>, <i>w</i>) = <i>f</i>(<i>v</i>)<i>g</i>(<i>w</i>) defines a bilinear map <i>V</i> × <i>W</i> → <i>F</i>.</li>
<li>The <a href="http://buttnet.org/wiki/Cross_product" title="Cross product">cross product</a> in <b>R</b><sup>3</sup> is a bilinear map <b>R</b><sup>3</sup> × <b>R</b><sup>3</sup> → <b>R</b><sup>3</sup>.</li>
<li>Let <i>B</i>&#160;: <i>V</i> × <i>W</i> → <i>X</i> be a bilinear map, and <i>L</i>&#160;: <i>U</i> → <i>W</i> be a <a href="http://buttnet.org/wiki/Linear_operator" title="Linear operator" class="mw-redirect">linear operator</a>, then (<i>v</i>, <i>u</i>) → <i>B</i>(<i>v</i>, <i>Lu</i>) is a bilinear map on <i>V</i> × <i>U</i></li>
<li>The <a href="http://buttnet.org/wiki/Zero_function" title="Zero function" class="mw-redirect">null map</a>, defined by <span class="texhtml"><i>B</i>(<i>v</i>,<i>w</i>) = <i>o</i></span> for all (<i>v</i>,<i>w</i>) in <i>V</i>×<i>W</i> is the only map from <i>V</i>×<i>W</i> to <i>X</i> which is bilinear and linear at the same time. Indeed, if (<i>v,w</i>)∈<i>V</i>×<i>W</i>, then if <i>B</i> is linear, <span class="texhtml"><i>B</i>(<i>v</i>,<i>w</i>) = <i>B</i>(<i>v</i>,<i>o</i>) + <i>B</i>(<i>o</i>,<i>w</i>) = <i>o</i> + <i>o</i></span> if <i>B</i> is bilinear.</li>
</ul>
<p><a name="See_also" id="See_also"></a></p>
<h2><span class="editsection">[<a href="http://buttnet.org/w/index.php?title=Bilinear_map&amp;action=edit&amp;section=4" title="Edit section: See also">edit</a>]</span> <span class="mw-headline">See also</span></h2>
<ul>
<li><a href="http://buttnet.org/wiki/Tensor_product" title="Tensor product">Tensor product</a></li>
<li><a href="http://buttnet.org/wiki/Multilinear_map" title="Multilinear map">Multilinear map</a></li>
<li><a href="http://buttnet.org/wiki/Sesquilinear_form" title="Sesquilinear form">Sesquilinear form</a></li>
<li><a href="http://buttnet.org/wiki/Bilinear_filtering" title="Bilinear filtering">Bilinear filtering</a></li>
</ul>
<p><a name="External_links" id="External_links"></a></p>
<h2><span class="editsection">[<a href="http://buttnet.org/w/index.php?title=Bilinear_map&amp;action=edit&amp;section=5" title="Edit section: External links">edit</a>]</span> <span class="mw-headline">External links</span></h2>
<ul>
<li><a href="http://www.umiacs.umd.edu/partnerships/lts/LTS_Report_Jan04.pdf" class="external text" title="http://www.umiacs.umd.edu/partnerships/lts/LTS_Report_Jan04.pdf" rel="nofollow">Use of Bilinear maps in cryptography</a> in <a href="http://wikileaks.org/wiki/On_the_take_and_loving_it" class="external text" title="http://wikileaks.org/wiki/On_the_take_and_loving_it" rel="nofollow">NSA sponsored academic research</a></li>
</ul>


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