Mike's lecture on vector spaces and dual spaces
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=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 widget that lets you plot vectors: [[media type="custom" key="15537326"]] 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
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<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 widget that lets you plot vectors:<br /> <br /> <!-- ws:start:WikiTextMediaRule:0:<img src="http://www.wikispaces.com/site/embedthumbnail/custom/15537326?h=0&w=0" class="WikiMedia WikiMediaCustom" id="wikitext@@media@@type=&quot;custom&quot; key=&quot;15537326&quot;" title="Custom Media"/> --><script type="text/javascript" id="WolframAlphaScriptf5af8de6802460753a75a4692d255641" src="http://www.wolframalpha.com/widget/widget.jsp?id=f5af8de6802460753a75a4692d255641&output=lightbox"> </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>