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

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-04-27 10:30:20 UTC</tt>.<br>

: The original revision id was <tt>326022510</tt>.<br>

<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__=

=[[media type="custom" key="15539114"]]=  

&lt;range type="comment" id="924583"&gt;$$\newcommand{\bra}[1]{\left \langle #1 \right |}

\newcommand{\s}[0]{\;\,}$$&lt;/range id="924583"&gt;

 

<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Mike's Lecture on Vector Spaces and Dual Spaces&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="LECTURE 1: Vector Spaces and Dual Spaces"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;&lt;u&gt;LECTURE 1: Vector Spaces and Dual Spaces&lt;/u&gt;&lt;/h1&gt;

&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;&lt;!-- ws:start:WikiTextMediaRule:0:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/15539114?h=0&amp;amp;w=0&amp;quot; class=&amp;quot;WikiMedia WikiMediaCustom&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;custom&amp;amp;quot; key=&amp;amp;quot;15539114&amp;amp;quot;&amp;quot; title=&amp;quot;Custom Media&amp;quot;/&amp;gt; --&gt;&lt;script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"&gt;

  &lt;!-- ws:start:WikiTextTocRule:16:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#LECTURE 1: Vector Spaces and Dual Spaces"&gt;LECTURE 1: Vector Spaces and Dual Spaces&lt;/a&gt;&lt;/div&gt;

&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#toc1"&gt;&lt;/a&gt;&lt;/div&gt;

&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#LECTURE 1: Vector Spaces and Dual Spaces-1.1: A monzo can be viewed as a VECTOR** in a **VECTOR SPACE."&gt;1.1: A monzo can be viewed as a VECTOR** in a **VECTOR SPACE.&lt;/a&gt;&lt;/div&gt;

&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#LECTURE 1: Vector Spaces and Dual Spaces-1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)"&gt;1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)&lt;/a&gt;&lt;/div&gt;

&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#LECTURE 1: Vector Spaces and Dual Spaces-1.3: Why the fact that covectors mean stuff matters. (OR: PREPARE FOR WEDGIE)"&gt;1.3: Why the fact that covectors mean stuff matters. (OR: PREPARE FOR WEDGIE)&lt;/a&gt;&lt;/div&gt;

&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt;&lt;/div&gt;

&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;a name="comment-924583-open" class="range"&gt;&lt;/a&gt;$$\newcommand{\bra}[1]{\left \langle #1 \right |}&lt;br /&gt;

\newcommand{\s}[0]{\;\,}$$&lt;a name="comment-924583-close" class="range"&gt;&lt;/a&gt;&lt;br /&gt;

&lt;br /&gt;

&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;

&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="LECTURE 1: Vector Spaces and Dual Spaces-1.1: A monzo can be viewed as a VECTOR** in a **VECTOR SPACE."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;1.1: A monzo can be viewed as a &lt;strong&gt;VECTOR&lt;/strong&gt; in a &lt;strong&gt;VECTOR SPACE&lt;/strong&gt;.&lt;/h2&gt;

&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="LECTURE 1: Vector Spaces and Dual Spaces-1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)&lt;/h2&gt;

&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="LECTURE 1: Vector Spaces and Dual Spaces-1.3: Why the fact that covectors mean stuff matters. (OR: PREPARE FOR WEDGIE)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;1.3: Why the fact that covectors mean stuff matters. (OR: PREPARE FOR WEDGIE)&lt;/h2&gt;

&lt;span style="font-size: 90%; vertical-align: sub;"&gt;&lt;!-- ws:start:WikiTextAnchorRule:23:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@ref1&amp;quot; title=&amp;quot;Anchor: ref1&amp;quot;/&amp;gt; --&gt;&lt;a name="ref1"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:23 --&gt;[1] - Note that some have raised technical concerns about this operation being called the &amp;quot;dot product,&amp;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 &amp;quot;&lt;strong&gt;bracket product&lt;/strong&gt;&amp;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.&lt;/span&gt;&lt;br /&gt;

&lt;span style="font-size: 90%;"&gt;&lt;!-- ws:start:WikiTextAnchorRule:24:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@ref2&amp;quot; title=&amp;quot;Anchor: ref2&amp;quot;/&amp;gt; --&gt;&lt;a name="ref2"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:24 --&gt;[2] - As we'll soon see, vals aren't the only sorts of covectors there are. There's another way to interpret the space of covectors, which is as &amp;quot;&lt;strong&gt;tuning maps&lt;/strong&gt;&amp;quot; which assign an actual tuning value in cents to the intervals in a temperament. But we're not there yet, so all you need to concern yourself with at this point is vals as covectors and monzos as vectors.&lt;/span&gt;&lt;br /&gt;

&lt;span style="font-size: 90%;"&gt;&lt;!-- ws:start:WikiTextAnchorRule:25:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@ref3&amp;quot; title=&amp;quot;Anchor: ref3&amp;quot;/&amp;gt; --&gt;&lt;a name="ref3"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:25 --&gt;[3] - This is one of those things that seems totally useless unless you're &lt;a class="wiki_link" href="/Gene%20Ward%20Smith"&gt;Gene Smith&lt;/a&gt;, at which point you realize that multivals represent higher-rank temperaments, and that this random field of mathematics has a rather musical interpretation.&lt;/span&gt;&lt;br /&gt;