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 06: | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-04-27 06:53:19 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>325952878</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> | ||
<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;white-space: pre-wrap ! important" class="old-revision-html">=LECTURE 1: Vector Spaces and Dual Spaces= | <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;white-space: pre-wrap ! important" class="old-revision-html">=[[media type="custom" key="15537538"]]= | ||
=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 haven't seen monzos before and are totally confused, please read the pages on [[xenharmonic/Monzos|Monzos]] and [[xenharmonic/Vals|Vals]] first! | ||
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Before we go on, however, let's clean up the notation a bit. In physics, the notation commonly used is to notate covectors <like this| and to notate vectors |like this>. Physicists call this "bra-ket" notation, or sometimes "Dirac" notation. So instead of writing covectors as (x, y, z)*, I'll just write <x y z| from now on. And instead of writing vectors as (a, b, c), I'll just write |a b c> from now on. | Before we go on, however, let's clean up the notation a bit. In physics, the notation commonly used is to notate covectors <like this| and to notate vectors |like this>. Physicists call this "bra-ket" notation, or sometimes "Dirac" notation. So instead of writing covectors as (x, y, z)*, I'll just write <x y z| from now on. And instead of writing vectors as (a, b, c), I'll just write |a b c> from now on. | ||
Technically, the application of <x y z| to |a b c> isn't called the dot product, for obscure mathematical reasons. It's sometimes called the "bracket product." But I've seen even Gene call it the "dot product" before, so I'm just going to informally use that usage for now because it's something everyone's familiar with (and it's basically the same exact thing). | Technically, the application of <x y z| to |a b c> isn't called the dot product, for obscure mathematical reasons. It's sometimes called the "bracket product." But I've seen even Gene call it the "dot product" before, so I'm just going to informally use that usage for now because it's something everyone's familiar with (and it's basically the same exact thing). | ||
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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> | ||
<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:4:&lt;h1&gt; --><h1 id="toc0"><a name="LECTURE 1: Vector Spaces and Dual Spaces"></a><!-- ws:end: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:4:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:4 --><!-- ws:start:WikiTextMediaRule:0:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/15537538?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;15537538&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; --><script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"> | ||
</script><!-- ws:end:WikiTextMediaRule:0 --></h1> | |||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc1"><a name="LECTURE 1: Vector Spaces and Dual Spaces"></a><!-- ws:end:WikiTextHeadingRule:6 -->LECTURE 1: Vector Spaces and Dual Spaces</h1> | |||
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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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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc2"><a name="LECTURE 1: Vector Spaces and Dual Spaces-1.1: A monzo can be viewed as a VECTOR** in a **VECTOR SPACE."></a><!-- ws:end:WikiTextHeadingRule:8 -->1.1: A monzo can be viewed as a <strong>VECTOR</strong> in a <strong>VECTOR SPACE</strong>.</h2> | ||
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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 widget that lets you plot vectors:<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 widget that lets you plot vectors:<br /> | ||
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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 &quot;acts on&quot; a vector, the interaction occurs by you taking the <strong>dot product</strong> of the two vectors.<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 &quot;acts on&quot; a vector, the interaction occurs by you taking the <strong>dot product</strong> of the two vectors.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h6&gt; --><h6 id="toc3"><a name="LECTURE 1: Vector Spaces and Dual Spaces-1.1: A monzo can 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:10 -->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:3:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/15537428?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;15537428&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; --><a href="#ref1" rel="nofollow"><sup>[1]</sup></a><!-- ws:end:WikiTextMediaRule:3 --> 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> | ||
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The action of a covector on a vector must, of course, be pictured as the different colored 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.<br /> | The action of a covector on a vector must, of course, be pictured as the different colored 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.<br /> | ||
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OK, so how do we use these things?<br /> | OK, so how do we use these things?<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc4"><a name="LECTURE 1: Vector Spaces and Dual Spaces-1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)"></a><!-- ws:end:WikiTextHeadingRule:12 -->1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)</h2> | ||
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One interesting way to think of covectors, since they're these dual vectors that &quot;act on&quot; 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 /> | One interesting way to think of covectors, since they're these dual vectors that &quot;act on&quot; 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 /> | ||
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Before we go on, however, let's clean up the notation a bit. In physics, the notation commonly used is to notate covectors &lt;like this| and to notate vectors |like this&gt;. Physicists call this &quot;bra-ket&quot; notation, or sometimes &quot;Dirac&quot; notation. So instead of writing covectors as (x, y, z)*, I'll just write &lt;x y z| from now on. And instead of writing vectors as (a, b, c), I'll just write |a b c&gt; from now on.<br /> | Before we go on, however, let's clean up the notation a bit. In physics, the notation commonly used is to notate covectors &lt;like this| and to notate vectors |like this&gt;. Physicists call this &quot;bra-ket&quot; notation, or sometimes &quot;Dirac&quot; notation. So instead of writing covectors as (x, y, z)*, I'll just write &lt;x y z| from now on. And instead of writing vectors as (a, b, c), I'll just write |a b c&gt; from now on.<br /> | ||
<br /> | <br /> | ||
Technically, the application of &lt;x y z| to |a b c&gt; isn't called the dot product, for obscure mathematical reasons. It's sometimes called the &quot;bracket product.&quot; But I've seen even Gene call it the &quot;dot product&quot; before, so I'm just going to informally use that usage for now because it's something everyone's familiar with (and it's basically the same exact thing).<br /> | Technically, the application of &lt;x y z| to |a b c&gt; isn't called the dot product, for obscure mathematical reasons. It's sometimes called the &quot;bracket product.&quot; But I've seen even Gene call it the &quot;dot product&quot; before, so I'm just going to informally use that usage for now because it's something everyone's familiar with (and it's basically the same exact thing).<br /> | ||
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For some of you, this may be review, but it's meant to give a basic foundation of the mathematical reasoning underpinning some of these objects. Stay tuned for more...<br /> | For some of you, this may be review, but it's meant to give a basic foundation of the mathematical reasoning underpinning some of these objects. Stay tuned for more...<br /> | ||
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<!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextAnchorRule:14:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@ref1&quot; title=&quot;Anchor: ref1&quot;/&gt; --><a name="ref1"></a><!-- ws:end:WikiTextAnchorRule:14 -->[1] - Note that some have raised technical concerns about this operation being called the &quot;dot product,&quot; 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 &quot;<strong>bracket product</strong>&quot;, 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></pre></div> | ||