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Mike's lecture on vector spaces and dual spaces: Difference between revisions

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**Imported revision 325960024 - Original comment: **
Wikispaces>mbattaglia1
**Imported revision 325960092 - Original comment: **
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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 07:30:26 UTC</tt>.<br>
: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-04-27 07:30:45 UTC</tt>.<br>
: The original revision id was <tt>325960024</tt>.<br>
: The original revision id was <tt>325960092</tt>.<br>
: The revision comment was: <tt></tt><br>
: The revision comment was: <tt></tt><br>
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>
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&lt;span style="display: block; text-align: center;"&gt;&lt;span class="MathJax"&gt;&lt;span class="math"&gt;&lt;span style="clip: rect(1.72em 1000em 2.742em -0.558em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.538em; width: 1.731em;"&gt;&lt;span class="mrow"&gt;&lt;span class="mi" style="font-family: MathJax_Math;"&gt;//test//&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;span style="display: block; text-align: center;"&gt;&lt;span class="MathJax"&gt;&lt;span class="math"&gt;&lt;span style="clip: rect(1.72em 1000em 2.742em -0.558em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.538em; width: 1.731em;"&gt;&lt;span class="mrow"&gt;&lt;span class="mi" style="font-family: MathJax_Math;"&gt;//test//&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;


[[media type="custom" key="15537772"]]
[[media type="custom" key="15537804"]]


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="15537774"]], where a b c are the exponents for primes 2, 3, and 5, respectively. A 7-limit JI monzo looks like [[media type="custom" key="15537776"]], where d represents the additional exponent for 7. The 11-limit gets you another coefficient and so on.
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="15537806"]], where a b c are the exponents for primes 2, 3, and 5, respectively. A 7-limit JI monzo looks like [[media type="custom" key="15537776"]], 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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  &lt;span style="display: block; text-align: center;"&gt;&lt;span class="MathJax"&gt;&lt;span class="math"&gt;&lt;span style="clip: rect(1.72em 1000em 2.742em -0.558em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.538em; width: 1.731em;"&gt;&lt;span class="mrow"&gt;&lt;span style="font-family: MathJax_Math;" class="mi"&gt;&lt;em&gt;test&lt;/em&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
  &lt;span style="display: block; text-align: center;"&gt;&lt;span class="MathJax"&gt;&lt;span class="math"&gt;&lt;span style="clip: rect(1.72em 1000em 2.742em -0.558em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.538em; width: 1.731em;"&gt;&lt;span class="mrow"&gt;&lt;span style="font-family: MathJax_Math;" class="mi"&gt;&lt;em&gt;test&lt;/em&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextMediaRule:1:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/15537772?h=0&amp;amp;w=0&amp;quot; class=&amp;quot;WikiMedia WikiMediaCustom&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;custom&amp;amp;quot; key=&amp;amp;quot;15537772&amp;amp;quot;&amp;quot; title=&amp;quot;Custom Media&amp;quot;/&amp;gt; --&gt;$$ \newcommand{\bra}[1]{\left \langle #1 \right |} \newcommand{\ket}[1]{\left |#1 \right \rangle} \newcommand{\braket}[2]{\left \langle #1 \middle |#2 \right \rangle}$$ \(\braket{a b c}{d e f}\)&lt;!-- ws:end:WikiTextMediaRule:1 --&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextMediaRule:1:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/15537804?h=0&amp;amp;w=0&amp;quot; class=&amp;quot;WikiMedia WikiMediaCustom&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;custom&amp;amp;quot; key=&amp;amp;quot;15537804&amp;amp;quot;&amp;quot; title=&amp;quot;Custom Media&amp;quot;/&amp;gt; --&gt;$$ \newcommand{\bra}[1]{\left \langle #1 \right |} \newcommand{\ket}[1]{\left |#1 \right \rangle} \newcommand{\braket}[2]{\left \langle #1 \middle |#2 \right \rangle}$$&lt;!-- ws:end:WikiTextMediaRule:1 --&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
If you haven't seen monzos or vals before and are totally confused, please read the pages on &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Monzos"&gt;Monzos&lt;/a&gt; and &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Vals"&gt;Vals&lt;/a&gt; first!&lt;br /&gt;
If you haven't seen monzos or vals before and are totally confused, please read the pages on &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Monzos"&gt;Monzos&lt;/a&gt; and &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Vals"&gt;Vals&lt;/a&gt; first!&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
If you have, then to review, a &lt;strong&gt;monzo&lt;/strong&gt; is a way to represent a JI interval that shows how it decomposes into a combination of simpler, &amp;quot;prime&amp;quot; intervals. It does so by directly representing an interval's prime factorization. A 5-limit monzo looks like &lt;!-- ws:start:WikiTextMediaRule:2:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/15537774?h=0&amp;amp;w=0&amp;quot; class=&amp;quot;WikiMedia WikiMediaCustom&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;custom&amp;amp;quot; key=&amp;amp;quot;15537774&amp;amp;quot;&amp;quot; title=&amp;quot;Custom Media&amp;quot;/&amp;gt; --&gt;\ket{a b c}&lt;!-- ws:end:WikiTextMediaRule:2 --&gt;, where a b c are the exponents for primes 2, 3, and 5, respectively. A 7-limit JI monzo looks like &lt;!-- ws:start:WikiTextMediaRule:3:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/15537776?h=0&amp;amp;w=0&amp;quot; class=&amp;quot;WikiMedia WikiMediaCustom&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;custom&amp;amp;quot; key=&amp;amp;quot;15537776&amp;amp;quot;&amp;quot; title=&amp;quot;Custom Media&amp;quot;/&amp;gt; --&gt;\ket{a b c d}&lt;!-- ws:end:WikiTextMediaRule:3 --&gt;, where d represents the additional exponent for 7. The 11-limit gets you another coefficient and so on.&lt;br /&gt;
If you have, then to review, a &lt;strong&gt;monzo&lt;/strong&gt; is a way to represent a JI interval that shows how it decomposes into a combination of simpler, &amp;quot;prime&amp;quot; intervals. It does so by directly representing an interval's prime factorization. A 5-limit monzo looks like &lt;!-- ws:start:WikiTextMediaRule:2:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/15537806?h=0&amp;amp;w=0&amp;quot; class=&amp;quot;WikiMedia WikiMediaCustom&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;custom&amp;amp;quot; key=&amp;amp;quot;15537806&amp;amp;quot;&amp;quot; title=&amp;quot;Custom Media&amp;quot;/&amp;gt; --&gt;\ket{\text{a b c}}&lt;!-- ws:end:WikiTextMediaRule:2 --&gt;, where a b c are the exponents for primes 2, 3, and 5, respectively. A 7-limit JI monzo looks like &lt;!-- ws:start:WikiTextMediaRule:3:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/15537776?h=0&amp;amp;w=0&amp;quot; class=&amp;quot;WikiMedia WikiMediaCustom&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;custom&amp;amp;quot; key=&amp;amp;quot;15537776&amp;amp;quot;&amp;quot; title=&amp;quot;Custom Media&amp;quot;/&amp;gt; --&gt;\ket{a b c d}&lt;!-- ws:end:WikiTextMediaRule:3 --&gt;, where d represents the additional exponent for 7. The 11-limit gets you another coefficient and so on.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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:&lt;br /&gt;
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:&lt;br /&gt;
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