Mike's lecture on vector spaces and dual spaces: Difference between revisions
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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 07: | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-04-27 07:37:58 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>325961558</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="15537816"]], where | 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="15537816"]], 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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[[#ref1]][1] - Note that some have raised technical concerns about this operation being called the "dot product," insisting that the dot product is something that's only done between two vectors, or two covectors, but never between one covector and one vector. Another term that's sometimes been used for this product in the "**bracket product**", for reasons we don't need to get into here. However, confusingly, the term bracket product has also been used for the ordinary dot product, and it's also very common to hear people call the thing I'm calling the dot product above. It's best at this point to just know that the two terms are out there. I'm going to continue calling it the dot product since its' something more people are familiar with.</pre></div> | [[#ref1]][1] - Note that some have raised technical concerns about this operation being called the "dot product," insisting that the dot product is something that's only done between two vectors, or two covectors, but never between one covector and one vector. Another term that's sometimes been used for this product in the "**bracket product**", for reasons we don't need to get into here. However, confusingly, the term bracket product has also been used for the ordinary dot product, and it's also very common to hear people call the thing I'm calling the dot product above. It's best at this point to just know that the two terms are out there. I'm going to continue calling it the dot product since its' something more people are familiar with.</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
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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/15537816?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;15537816&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; -->\(\ket{\text{a b c}}\)<!-- ws:end:WikiTextMediaRule:1 -->, where | 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/15537816?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;15537816&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; -->\(\ket{\text{a b c}}\)<!-- 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 /> | ||