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

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**Imported revision 326015976 - Original comment: **
Wikispaces>mbattaglia1
**Imported revision 326016478 - Original comment: **
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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 10:14:49 UTC</tt>.<br>
: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-04-27 10:16:06 UTC</tt>.<br>
: The original revision id was <tt>326015976</tt>.<br>
: The original revision id was <tt>326016478</tt>.<br>
: The revision comment was: <tt></tt><br>
: The revision comment was: <tt></tt><br>
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">=[[media type="custom" key="15539114"]]=  
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[[toc]]
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[[toc]]


=__LECTURE 1: Vector Spaces and Dual Spaces__=  
=__LECTURE 1: Vector Spaces and Dual Spaces__=  
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OK, so how do we use these things?
OK, so how do we use these things?


&lt;span style="color: #aaaaaa;"&gt;
 
&lt;/span&gt;
 
----
----
==1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)==  
==1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)==  
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But where do we go with this realization?
But where do we go with this realization?


&lt;span style="color: #aaaaaa;"&gt;
 
&lt;/span&gt;
 
----
----
==1.3: Why the fact that covectors mean stuff matters. (OR: PREPARE FOR WEDGIE)==  
==1.3: Why the fact that covectors mean stuff matters. (OR: PREPARE FOR WEDGIE)==  
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//]]&gt;
&lt;/script&gt;&lt;!-- ws:end:WikiTextMediaRule:0 --&gt;&lt;/h1&gt;
&lt;/script&gt;&lt;!-- ws:end:WikiTextMediaRule:0 --&gt;&lt;/h1&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="#toc0"&gt;&lt;/a&gt;&lt;/div&gt;
  $$&lt;br /&gt;
&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&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: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: 6em;"&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.----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{1} of the two is \(12 \cdot \-4 + 19 \cdot 4 + 28 \cdot \-1 = 0\). Thus, the result of \((12,19,28)^*\) acting on \((\-4,4,\-1)\) is \(0\)."&gt;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{1} of the two is \(12 \cdot \-4 + 19 \cdot 4 + 28 \cdot \-1 = 0\). Thus, the result of \((12,19,28)^*\) acting on \((\-4,4,\-1)\) is \(0\).&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-1.2: 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:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&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:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:23 --&gt;$$&lt;br /&gt;
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&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="#toc0"&gt;&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="#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: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: 6em;"&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.----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{1} of the two is \(12 \cdot \-4 + 19 \cdot 4 + 28 \cdot \-1 = 0\). Thus, the result of \((12,19,28)^*\) acting on \((\-4,4,\-1)\) is \(0\)."&gt;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{1} of the two is \(12 \cdot \-4 + 19 \cdot 4 + 28 \cdot \-1 = 0\). Thus, the result of \((12,19,28)^*\) acting on \((\-4,4,\-1)\) is \(0\).&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-1.2: 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:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&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:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt;&lt;/div&gt;
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&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&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:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&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;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;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;
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OK, so how do we use these things?&lt;br /&gt;
OK, so how do we use these things?&lt;br /&gt;
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&lt;span style="color: #aaaaaa;"&gt;&lt;br /&gt;
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&lt;/span&gt;&lt;br /&gt;
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&lt;hr /&gt;
&lt;hr /&gt;
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&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&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;
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But where do we go with this realization?&lt;br /&gt;
But where do we go with this realization?&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;span style="color: #aaaaaa;"&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;/span&gt;&lt;br /&gt;
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&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&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;
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