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

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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;

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!

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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;

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;

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For instance, the syntonic comma is \(\ket{\-4 \s 4 \s \-1}\). 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:&lt;br /&gt;

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One interesting way to think of covectors, since they're these dual vectors that &amp;quot;act on&amp;quot; normal vectors, is 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.&lt;br /&gt;

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Assuming you've understood my exposition thus far, you now hopefully see where all things like monzos and vals come from.&lt;br /&gt;