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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 | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-04-27 05:30:10 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>325936154</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> | ||
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For instance, the syntonic comma is |-4 4 -1>. A geometric interpretation of this interval might be as a point in a space, like the point (-4, 4, -1). You'd plot this point by going -4 steps on the x axis, 4 steps on the y axis, and -1 steps on the z-axis. And if you really want to think of it like a vector in the sense that some high school or college algebra courses teach it, you can also draw an arrow with a big arrowhead from the origin that connects to this point. Here's a picture of that: | For instance, the syntonic comma is |-4 4 -1>. A geometric interpretation of this interval might be as a point in a space, like the point (-4, 4, -1). You'd plot this point by going -4 steps on the x axis, 4 steps on the y axis, and -1 steps on the z-axis. And if you really want to think of it like a vector in the sense that some high school or college algebra courses teach it, you can also draw an arrow with a big arrowhead from the origin that connects to this point. Here's a picture of that: | ||
[[media type="custom" key="15536954"]] | |||
Paul's "A Middle Path" paper has so many good plots of this that I might as well just point anyone interested to take a look at it over there: [[http://sethares.engr.wisc.edu/paperspdf/Erlich-MiddlePath.pdf]] | |||
Now, here's the interesting part: in linear algebra, every vector space has a "dual space," which of course must be thought of as a bizarro universe for the vector space in which the background is black and the arrows and points are white. The elements in this space are called "covectors." I can't get the exact colors I mentioned here, but I've cheated a bit to get Wolfram to change the colors, so you can plot covectors here: | |||
Covectors can "interact" with vectors, or rather "act on" them, by taking the dot product of the covector and a vector. So for instance, if your covector is (12, 19, 28)* (the star means it's in the dual space), and your vector is (-4, 4, -1), then the dot product of the two is 12*-4 + 19*4 * 28*-1 = 0. This must of course be pictured as the black and white arrows lining up and exploding and spitting out a single number, or something. | Covectors can "interact" with vectors, or rather "act on" them, by taking the dot product of the covector and a vector. So for instance, if your covector is (12, 19, 28)* (the star means it's in the dual space), and your vector is (-4, 4, -1), then the dot product of the two is 12*-4 + 19*4 * 28*-1 = 0. This must of course be pictured as the black and white arrows lining up and exploding and spitting out a single number, or something. | ||
In a drier sense, a covector can also be thought of as a type of function that takes in a vector and spits out a number. So (12, 19, 28)* can also be thought of as f(v) = 12a + 19b + 28c for some vector of the form (a, b, c). | In a drier sense, a covector can also be thought of as a type of function that takes in a vector and spits out a number. So (12, 19, 28)* can also be thought of as f(v) = 12a + 19b + 28c for some vector of the form (a, b, c). | ||
OK, what the hell does all of that mean? END LESSON 1</pre></div> | OK, what the hell does all of that mean? END LESSON 1</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: | <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:1:&lt;h1&gt; --><h1 id="toc0"><a name="LECTURE 1: Vector Spaces and Dual Spaces"></a><!-- ws:end:WikiTextHeadingRule:1 -->LECTURE 1: Vector Spaces and Dual Spaces</h1> | ||
<br /> | <br /> | ||
If you haven't seen monzos 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 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 /> | ||
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For instance, the syntonic comma is |-4 4 -1&gt;. A geometric interpretation of this interval might be as a point in a space, like the point (-4, 4, -1). You'd plot this point by going -4 steps on the x axis, 4 steps on the y axis, and -1 steps on the z-axis. And if you really want to think of it like a vector in the sense that some high school or college algebra courses teach it, you can also draw an arrow with a big arrowhead from the origin that connects to this point. Here's a picture of that:<br /> | For instance, the syntonic comma is |-4 4 -1&gt;. A geometric interpretation of this interval might be as a point in a space, like the point (-4, 4, -1). You'd plot this point by going -4 steps on the x axis, 4 steps on the y axis, and -1 steps on the z-axis. And if you really want to think of it like a vector in the sense that some high school or college algebra courses teach it, you can also draw an arrow with a big arrowhead from the origin that connects to this point. Here's a picture of that:<br /> | ||
<br /> | <br /> | ||
< | <!-- ws:start:WikiTextMediaRule:0:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/15536954?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;15536954&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; --><script type="text/javascript" id="WolframAlphaScript207d0ac77d88b7c7f3b28a5f309715d" src="http://www.wolframalpha.com/widget/widget.jsp?id=207d0ac77d88b7c7f3b28a5f309715d&amp;theme=orange"> | ||
</script><!-- ws:end:WikiTextMediaRule:0 --><br /> | |||
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Paul's &quot;A Middle Path&quot; paper has so many good plots of this that I might as well just point anyone interested to take a look at it over there: <a class="wiki_link_ext" href="http://sethares.engr.wisc.edu/paperspdf/Erlich-MiddlePath.pdf" rel="nofollow">http://sethares.engr.wisc.edu/paperspdf/Erlich-MiddlePath.pdf</a><br /> | |||
<br /> | |||
Now, here's the interesting part: in linear algebra, every vector space has a &quot;dual space,&quot; which of course must be thought of as a bizarro universe for the vector space in which the background is black and the arrows and points are white. The elements in this space are called &quot;covectors.&quot; I can't get the exact colors I mentioned here, but I've cheated a bit to get Wolfram to change the colors, so you can plot covectors here:<br /> | |||
<br /> | |||
<br /> | <br /> | ||
Covectors can &quot;interact&quot; with vectors, or rather &quot;act on&quot; them, by taking the dot product of the covector and a vector. So for instance, if your covector is (12, 19, 28)* (the star means it's in the dual space), and your vector is (-4, 4, -1), then the dot product of the two is 12*-4 + 19*4 * 28*-1 = 0. This must of course be pictured as the black and white arrows lining up and exploding and spitting out a single number, or something.<br /> | Covectors can &quot;interact&quot; with vectors, or rather &quot;act on&quot; them, by taking the dot product of the covector and a vector. So for instance, if your covector is (12, 19, 28)* (the star means it's in the dual space), and your vector is (-4, 4, -1), then the dot product of the two is 12*-4 + 19*4 * 28*-1 = 0. This must of course be pictured as the black and white arrows lining up and exploding and spitting out a single number, or something.<br /> | ||
In a drier sense, a covector can also be thought of as a type of function that takes in a vector and spits out a number. So (12, 19, 28)* can also be thought of as f(v) = 12a + 19b + 28c for some vector of the form (a, b, c).<br /> | In a drier sense, a covector can also be thought of as a type of function that takes in a vector and spits out a number. So (12, 19, 28)* can also be thought of as f(v) = 12a + 19b + 28c for some vector of the form (a, b, c).<br /> | ||
OK, what the hell does all of that mean? END LESSON 1</body></html></pre></div> | OK, what the hell does all of that mean? END LESSON 1</body></html></pre></div> | ||
Revision as of 05:30, 27 April 2012
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author mbattaglia1 and made on 2012-04-27 05:30:10 UTC.
- The original revision id was 325936154.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
=LECTURE 1: Vector Spaces and Dual Spaces= If you haven't seen monzos 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 |a b c>, where a b c are the exponents for primes 2, 3, and 5, respectively. A 7-limit JI monzo looks like |a b c d>, 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: A monzo can also be viewed as a **VECTOR** in a **VECTOR SPACE**. For instance, the syntonic comma is |-4 4 -1>. A geometric interpretation of this interval might be as a point in a space, like the point (-4, 4, -1). You'd plot this point by going -4 steps on the x axis, 4 steps on the y axis, and -1 steps on the z-axis. And if you really want to think of it like a vector in the sense that some high school or college algebra courses teach it, you can also draw an arrow with a big arrowhead from the origin that connects to this point. Here's a picture of that: [[media type="custom" key="15536954"]] Paul's "A Middle Path" paper has so many good plots of this that I might as well just point anyone interested to take a look at it over there: [[http://sethares.engr.wisc.edu/paperspdf/Erlich-MiddlePath.pdf]] Now, here's the interesting part: in linear algebra, every vector space has a "dual space," which of course must be thought of as a bizarro universe for the vector space in which the background is black and the arrows and points are white. The elements in this space are called "covectors." I can't get the exact colors I mentioned here, but I've cheated a bit to get Wolfram to change the colors, so you can plot covectors here: Covectors can "interact" with vectors, or rather "act on" them, by taking the dot product of the covector and a vector. So for instance, if your covector is (12, 19, 28)* (the star means it's in the dual space), and your vector is (-4, 4, -1), then the dot product of the two is 12*-4 + 19*4 * 28*-1 = 0. This must of course be pictured as the black and white arrows lining up and exploding and spitting out a single number, or something. In a drier sense, a covector can also be thought of as a type of function that takes in a vector and spits out a number. So (12, 19, 28)* can also be thought of as f(v) = 12a + 19b + 28c for some vector of the form (a, b, c). OK, what the hell does all of that mean? END LESSON 1
Original HTML content:
<html><head><title>Mike's Lecture on Vector Spaces and Dual Spaces</title></head><body><!-- ws:start:WikiTextHeadingRule:1:<h1> --><h1 id="toc0"><a name="LECTURE 1: Vector Spaces and Dual Spaces"></a><!-- ws:end:WikiTextHeadingRule:1 -->LECTURE 1: Vector Spaces and Dual Spaces</h1> <br /> If you haven't seen monzos 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 /> 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, "prime" intervals. It does so by directly representing an interval's prime factorization. A 5-limit monzo looks like |a b c>, where a b c are the exponents for primes 2, 3, and 5, respectively. A 7-limit JI monzo looks like |a b c d>, where d represents the additional exponent for 7. The 11-limit gets you another coefficient and so on.<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 /> <br /> A monzo can also be viewed as a <strong>VECTOR</strong> in a <strong>VECTOR SPACE</strong>.<br /> <br /> For instance, the syntonic comma is |-4 4 -1>. A geometric interpretation of this interval might be as a point in a space, like the point (-4, 4, -1). You'd plot this point by going -4 steps on the x axis, 4 steps on the y axis, and -1 steps on the z-axis. And if you really want to think of it like a vector in the sense that some high school or college algebra courses teach it, you can also draw an arrow with a big arrowhead from the origin that connects to this point. Here's a picture of that:<br /> <br /> <!-- ws:start:WikiTextMediaRule:0:<img src="http://www.wikispaces.com/site/embedthumbnail/custom/15536954?h=0&w=0" class="WikiMedia WikiMediaCustom" id="wikitext@@media@@type=&quot;custom&quot; key=&quot;15536954&quot;" title="Custom Media"/> --><script type="text/javascript" id="WolframAlphaScript207d0ac77d88b7c7f3b28a5f309715d" src="http://www.wolframalpha.com/widget/widget.jsp?id=207d0ac77d88b7c7f3b28a5f309715d&theme=orange"> </script><!-- ws:end:WikiTextMediaRule:0 --><br /> <br /> Paul's "A Middle Path" paper has so many good plots of this that I might as well just point anyone interested to take a look at it over there: <a class="wiki_link_ext" href="http://sethares.engr.wisc.edu/paperspdf/Erlich-MiddlePath.pdf" rel="nofollow">http://sethares.engr.wisc.edu/paperspdf/Erlich-MiddlePath.pdf</a><br /> <br /> Now, here's the interesting part: in linear algebra, every vector space has a "dual space," which of course must be thought of as a bizarro universe for the vector space in which the background is black and the arrows and points are white. The elements in this space are called "covectors." I can't get the exact colors I mentioned here, but I've cheated a bit to get Wolfram to change the colors, so you can plot covectors here:<br /> <br /> <br /> Covectors can "interact" with vectors, or rather "act on" them, by taking the dot product of the covector and a vector. So for instance, if your covector is (12, 19, 28)* (the star means it's in the dual space), and your vector is (-4, 4, -1), then the dot product of the two is 12*-4 + 19*4 * 28*-1 = 0. This must of course be pictured as the black and white arrows lining up and exploding and spitting out a single number, or something.<br /> In a drier sense, a covector can also be thought of as a type of function that takes in a vector and spits out a number. So (12, 19, 28)* can also be thought of as f(v) = 12a + 19b + 28c for some vector of the form (a, b, c).<br /> OK, what the hell does all of that mean? END LESSON 1</body></html>