Mike's lecture on vector spaces and dual spaces: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 325945264 - Original comment: ** |
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Paul's "A Middle Path" paper has so many good plots of this that I might as well just point anyone interested to take a look at it over there: [[http://sethares.engr.wisc.edu/paperspdf/Erlich-MiddlePath.pdf]] | Keep in mind that Wolfram Alpha is very fragile, so if you try to do anything fancy, it's going to break. But, Paul's "A Middle Path" paper has so many good plots of this that I might as well just point anyone interested to take a look at it over there: [[http://sethares.engr.wisc.edu/paperspdf/Erlich-MiddlePath.pdf]] | ||
Now, here's the interesting part: in linear algebra, every vector space has a "dual space," which of course must be thought of as a bizarro universe for the vector space in which the background is black and the arrows and points are white. The elements in this space are called "covectors." I can't get the exact colors I mentioned here, but I've cheated a bit to get Wolfram to change the colors, so you can plot covectors here: | Now, here's the interesting part: in linear algebra, every vector space has a "dual space," which of course must be thought of as a bizarro universe for the vector space in which the background is black and the arrows and points are white. The elements in this space are called "covectors." I can't get the exact colors I mentioned here, but I've cheated a bit to get Wolfram to change the colors, so you can plot covectors here: | ||
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Paul's &quot;A Middle Path&quot; paper has so many good plots of this that I might as well just point anyone interested to take a look at it over there: <a class="wiki_link_ext" href="http://sethares.engr.wisc.edu/paperspdf/Erlich-MiddlePath.pdf" rel="nofollow">http://sethares.engr.wisc.edu/paperspdf/Erlich-MiddlePath.pdf</a><br /> | Keep in mind that Wolfram Alpha is very fragile, so if you try to do anything fancy, it's going to break. But, Paul's &quot;A Middle Path&quot; paper has so many good plots of this that I might as well just point anyone interested to take a look at it over there: <a class="wiki_link_ext" href="http://sethares.engr.wisc.edu/paperspdf/Erlich-MiddlePath.pdf" rel="nofollow">http://sethares.engr.wisc.edu/paperspdf/Erlich-MiddlePath.pdf</a><br /> | ||
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Now, here's the interesting part: in linear algebra, every vector space has a &quot;dual space,&quot; which of course must be thought of as a bizarro universe for the vector space in which the background is black and the arrows and points are white. The elements in this space are called &quot;covectors.&quot; I can't get the exact colors I mentioned here, but I've cheated a bit to get Wolfram to change the colors, so you can plot covectors here:<br /> | Now, here's the interesting part: in linear algebra, every vector space has a &quot;dual space,&quot; which of course must be thought of as a bizarro universe for the vector space in which the background is black and the arrows and points are white. The elements in this space are called &quot;covectors.&quot; I can't get the exact colors I mentioned here, but I've cheated a bit to get Wolfram to change the colors, so you can plot covectors here:<br /> | ||
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