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: ==1.1: 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"]] Keep in mind that Wolfram Alpha is very fragile, so if you try to do anything fancy, it's going to break. But, 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: [[media type="custom" key="15537360"]] You can also plot more than one vector or covector by putting in a list of vectors separated by commas, something like (12, 19, 28), (7, 11, 16). However, this will break the nice color properties I set up above. Also, if you put in too many entries, Wolfram Alpha has been known to break. **So then, what's the point?** This is all well and good by itself, but it doesn't mean anything unless you understand how covectors interact with vectors. Covectors are mathematical objects that are thought to //act on// vectors. When a covector "acts on" a vector, the interaction occurs by you taking the **dot product** of the two vectors. ======For example: say 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[[media type="custom" key="15537428"]] of the two is 12*-4 + 19*4 * 28*-1 = 0. Thus, the result of (12, 19, 28) acting on (-4, 4, -1) is 0.====== The action of a covector on a vector must, of course, be pictured as the different colored white arrows lining up and exploding and spitting out a single number, or something. Wolfram unfortunately doesn't let me do nice explosion effects, so you'll have to imagine it. Alternatively, if you're used to dot products, then you'll also be able to understand this operation in terms of the projection of one vector on another. However, if you choose to visualize it this way, you must understand that these two things you're projecting onto one another lie in __different__ spaces - one is a vector lying in a vector space, and the other is a covector lying in this new "dual space" you're learning about. One interesting way to think of covectors, since they're these dual vectors that "act on" normal vectors, is thus as functions - they take in a vector as input, multiply each coefficient of the vector by the corresponding coefficient of the covector, sum them up, and spit out a number. In other words, you know that the action of the covector (12, 19, 28)* on any arbitrary vector (a, b, c) is going to be 12a + 19b + 28c. So, you can think of (12, 19, 28)* itself as a function looking something like f(**v**) = 12a + 19b + 28c for some vector of the form (a, b, c). I've bolded the **v** in f(**v**) to specify that **v** is a vector that's being taken in as input. Thus, covectors are little mathematical machines - they take in vectors, do some simple dot product-ish stuff, and output a scalar. Simple! OK, how do we use these things then? [[#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.
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<html><head><title>Mike's Lecture on Vector Spaces and Dual Spaces</title></head><body><!-- ws:start:WikiTextHeadingRule:3:<h1> --><h1 id="toc0"><a name="LECTURE 1: Vector Spaces and Dual Spaces"></a><!-- ws:end:WikiTextHeadingRule:3 -->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 /> <br /> <!-- ws:start:WikiTextHeadingRule:5:<h2> --><h2 id="toc1"><a name="LECTURE 1: Vector Spaces and Dual Spaces-1.1: A monzo can also be viewed as a VECTOR** in a **VECTOR SPACE."></a><!-- ws:end:WikiTextHeadingRule:5 -->1.1: A monzo can also be viewed as a <strong>VECTOR</strong> in a <strong>VECTOR SPACE</strong>.</h2> <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 /> Keep in mind that Wolfram Alpha is very fragile, so if you try to do anything fancy, it's going to break. But, 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 /> <!-- ws:start:WikiTextMediaRule:1:<img src="http://www.wikispaces.com/site/embedthumbnail/custom/15537360?h=0&w=0" class="WikiMedia WikiMediaCustom" id="wikitext@@media@@type=&quot;custom&quot; key=&quot;15537360&quot;" title="Custom Media"/> --><script type="text/javascript" id="WolframAlphaScriptca79995ff5942e9f187c05cd2fce394b" src="http://www.wolframalpha.com/widget/widget.jsp?id=ca79995ff5942e9f187c05cd2fce394b"> </script><!-- ws:end:WikiTextMediaRule:1 --><br /> <br /> You can also plot more than one vector or covector by putting in a list of vectors separated by commas, something like (12, 19, 28), (7, 11, 16). However, this will break the nice color properties I set up above. Also, if you put in too many entries, Wolfram Alpha has been known to break.<br /> <br /> <strong>So then, what's the point?</strong><br /> <br /> This is all well and good by itself, but it doesn't mean anything unless you understand how covectors interact with vectors. Covectors are mathematical objects that are thought to <em>act on</em> vectors. When a covector "acts on" a vector, the interaction occurs by you taking the <strong>dot product</strong> of the two vectors.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:7:<h6> --><h6 id="toc2"><a name="LECTURE 1: Vector Spaces and Dual Spaces-1.1: A monzo can also be viewed as a VECTOR** in a **VECTOR SPACE.----For example: say 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. Thus, the result of (12, 19, 28) acting on (-4, 4, -1) is 0."></a><!-- ws:end:WikiTextHeadingRule:7 -->For example: say 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<!-- ws:start:WikiTextMediaRule:2:<img src="http://www.wikispaces.com/site/embedthumbnail/custom/15537428?h=0&w=0" class="WikiMedia WikiMediaCustom" id="wikitext@@media@@type=&quot;custom&quot; key=&quot;15537428&quot;" title="Custom Media"/> --><a href="#ref1" rel="nofollow"><sup>[1]</sup></a><!-- ws:end:WikiTextMediaRule:2 --> of the two is 12*-4 + 19*4 * 28*-1 = 0. Thus, the result of (12, 19, 28) acting on (-4, 4, -1) is 0.</h6> <br /> The action of a covector on a vector must, of course, be pictured as the different colored white arrows lining up and exploding and spitting out a single number, or something. Wolfram unfortunately doesn't let me do nice explosion effects, so you'll have to imagine it. Alternatively, if you're used to dot products, then you'll also be able to understand this operation in terms of the projection of one vector on another. However, if you choose to visualize it this way, you must understand that these two things you're projecting onto one another lie in <u>different</u> spaces - one is a vector lying in a vector space, and the other is a covector lying in this new "dual space" you're learning about.<br /> <br /> One interesting way to think of covectors, since they're these dual vectors that "act on" normal vectors, is thus as functions - they take in a vector as input, multiply each coefficient of the vector by the corresponding coefficient of the covector, sum them up, and spit out a number. In other words, you know that the action of the covector (12, 19, 28)* on any arbitrary vector (a, b, c) is going to be 12a + 19b + 28c. So, you can think of (12, 19, 28)* itself as a function looking something like f(<strong>v</strong>) = 12a + 19b + 28c for some vector of the form (a, b, c). I've bolded the <strong>v</strong> in f(<strong>v</strong>) to specify that <strong>v</strong> is a vector that's being taken in as input.<br /> <br /> Thus, covectors are little mathematical machines - they take in vectors, do some simple dot product-ish stuff, and output a scalar. Simple!<br /> <br /> OK, how do we use these things then?<br /> <br /> <br /> <br /> <!-- ws:start:WikiTextAnchorRule:9:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@ref1" title="Anchor: ref1"/> --><a name="ref1"></a><!-- ws:end:WikiTextAnchorRule:9 -->[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 "<strong>bracket product</strong>", 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.</body></html>