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

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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>
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: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-04-27 09:34:24 UTC</tt>.<br>
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: The original revision id was <tt>325998586</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>
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[[media type="custom" key="15537360"]]
[[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 \((12,19,28),\q(7,11,16)\). 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.
You can also plot more than one vector or covector by putting in a list of vectors separated by commas, something like \((12,19,28),\s(7,11,16)\). 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?**
**So then, what's the point?**
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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 //act on// vectors. When a covector "acts on" a vector, the interaction occurs by you taking the **dot product** of the two vectors.
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&lt;span style="font-size: 80%; vertical-align: super;"&gt;[[Mike's Lecture on Vector Spaces and Dual Spaces#ref1|{1}]]&lt;/span&gt; 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 \(0\).======  
======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&lt;span style="font-size: 80%; vertical-align: super;"&gt;[[Mike's Lecture on Vector Spaces and Dual Spaces#ref1|{1}]]&lt;/span&gt; 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 \(0\).======  


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.
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.
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&lt;/script&gt;&lt;!-- ws:end:WikiTextMediaRule:2 --&gt;&lt;br /&gt;
&lt;/script&gt;&lt;!-- ws:end:WikiTextMediaRule:2 --&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
You can also plot more than one vector or covector by putting in a list of vectors separated by commas, something like \((12,19,28),\q(7,11,16)\). 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.&lt;br /&gt;
You can also plot more than one vector or covector by putting in a list of vectors separated by commas, something like \((12,19,28),\s(7,11,16)\). 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.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;So then, what's the point?&lt;/strong&gt;&lt;br /&gt;
&lt;strong&gt;So then, what's the point?&lt;/strong&gt;&lt;br /&gt;
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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 &lt;em&gt;act on&lt;/em&gt; vectors. When a covector &amp;quot;acts on&amp;quot; a vector, the interaction occurs by you taking the &lt;strong&gt;dot product&lt;/strong&gt; of the two vectors.&lt;br /&gt;
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 &lt;em&gt;act on&lt;/em&gt; vectors. When a covector &amp;quot;acts on&amp;quot; a vector, the interaction occurs by you taking the &lt;strong&gt;dot product&lt;/strong&gt; of the two vectors.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h6&amp;gt; --&gt;&lt;h6 id="toc3"&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.----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;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&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&lt;span style="font-size: 80%; vertical-align: super;"&gt;&lt;a class="wiki_link" href="/Mike%27s%20Lecture%20on%20Vector%20Spaces%20and%20Dual%20Spaces#ref1"&gt;{1}&lt;/a&gt;&lt;/span&gt; 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;/h6&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h6&amp;gt; --&gt;&lt;h6 id="toc3"&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.----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;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&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&lt;span style="font-size: 80%; vertical-align: super;"&gt;&lt;a class="wiki_link" href="/Mike%27s%20Lecture%20on%20Vector%20Spaces%20and%20Dual%20Spaces#ref1"&gt;{1}&lt;/a&gt;&lt;/span&gt; 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;/h6&gt;
  &lt;br /&gt;
  &lt;br /&gt;
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.&lt;br /&gt;
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.&lt;br /&gt;
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