Mike's lecture on vector spaces and dual spaces: Difference between revisions
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-04-27 08: | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-04-27 08:04:59 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>325968020</tt>.<br> | ||
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
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=LECTURE 1: Vector Spaces and Dual Spaces= | =LECTURE 1: Vector Spaces and Dual Spaces= | ||
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If you haven't seen monzos or vals before and are totally confused, please read the pages on [[xenharmonic/Monzos|Monzos]] and [[xenharmonic/Vals|Vals]] first! | If you haven't seen monzos or vals before and are totally confused, please read the pages on [[xenharmonic/Monzos|Monzos]] and [[xenharmonic/Vals|Vals]] first! | ||
If you have, then to review, a **monzo** is a way to represent a JI interval that shows how it decomposes into a combination of simpler, "prime" intervals. It does so by directly representing an interval's prime factorization. A 5-limit monzo looks like [[media type="custom" key=" | If you have, then to review, a **monzo** is a way to represent a JI interval that shows how it decomposes into a combination of simpler, "prime" intervals. It does so by directly representing an interval's prime factorization. A 5-limit monzo looks like [[media type="custom" key="15538076"]], where a, b, and c are the exponents for primes 2, 3, and 5, respectively. A 7-limit JI monzo looks like [[media type="custom" key="15537820"]], where d represents the additional exponent for 7. The 11-limit gets you another coefficient and so on. | ||
Assuming you understand that, then we've reached our first new idea, which will help us gain a geometric intuition into what some of these abstract entities mean. That idea is this: | Assuming you understand that, then we've reached our first new idea, which will help us gain a geometric intuition into what some of these abstract entities mean. That idea is this: | ||
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[[media type="custom" key="15537974"]]</pre></div> | [[media type="custom" key="15537974"]]</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Mike's Lecture on Vector Spaces and Dual Spaces</title></head><body><!-- ws:start:WikiTextHeadingRule:9:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:9 --><!-- ws:start:WikiTextMediaRule:0:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/ | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Mike's Lecture on Vector Spaces and Dual Spaces</title></head><body><!-- ws:start:WikiTextHeadingRule:9:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:9 --><!-- ws:start:WikiTextMediaRule:0:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/15538084?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;15538084&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; --><script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"> | ||
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<!-- ws:start:WikiTextHeadingRule:11:&lt;h1&gt; --><h1 id="toc1"><a name="LECTURE 1: Vector Spaces and Dual Spaces"></a><!-- ws:end:WikiTextHeadingRule:11 -->LECTURE 1: Vector Spaces and Dual Spaces</h1> | <!-- ws:start:WikiTextHeadingRule:11:&lt;h1&gt; --><h1 id="toc1"><a name="LECTURE 1: Vector Spaces and Dual Spaces"></a><!-- ws:end:WikiTextHeadingRule:11 -->LECTURE 1: Vector Spaces and Dual Spaces</h1> | ||
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If you haven't seen monzos or vals before and are totally confused, please read the pages on <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Monzos">Monzos</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Vals">Vals</a> first!<br /> | If you haven't seen monzos or vals before and are totally confused, please read the pages on <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Monzos">Monzos</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Vals">Vals</a> first!<br /> | ||
<br /> | <br /> | ||
If you have, then to review, a <strong>monzo</strong> is a way to represent a JI interval that shows how it decomposes into a combination of simpler, &quot;prime&quot; intervals. It does so by directly representing an interval's prime factorization. A 5-limit monzo looks like <!-- ws:start:WikiTextMediaRule:1:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/ | If you have, then to review, a <strong>monzo</strong> is a way to represent a JI interval that shows how it decomposes into a combination of simpler, &quot;prime&quot; intervals. It does so by directly representing an interval's prime factorization. A 5-limit monzo looks like <!-- ws:start:WikiTextMediaRule:1:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/15538076?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;15538076&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; -->\(\kettext{a b c}\)<!-- ws:end:WikiTextMediaRule:1 -->, where a, b, and c are the exponents for primes 2, 3, and 5, respectively. A 7-limit JI monzo looks like <!-- ws:start:WikiTextMediaRule:2:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/15537820?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;15537820&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; -->\(\ket{\text{a b c d}}\)<!-- ws:end:WikiTextMediaRule:2 -->, where d represents the additional exponent for 7. The 11-limit gets you another coefficient and so on.<br /> | ||
<br /> | <br /> | ||
Assuming you understand that, then we've reached our first new idea, which will help us gain a geometric intuition into what some of these abstract entities mean. That idea is this:<br /> | Assuming you understand that, then we've reached our first new idea, which will help us gain a geometric intuition into what some of these abstract entities mean. That idea is this:<br /> | ||