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Directory:Jon Awbrey/Papers/Differential Logic (view source)

Revision as of 18:40, 11 June 2009

182 bytes added ,  18:40, 11 June 2009
→‎Note 23: format table of matrices
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These days one is much more likely to encounter the natural representation of <math>S_3\!</math> in the form of a ''linear representation'', that is, as a family of linear transformations that map the elements of a suitable vector space into each other, all of which would in turn usually be represented by a set of matrices like these:
 
These days one is much more likely to encounter the natural representation of <math>S_3\!</math> in the form of a ''linear representation'', that is, as a family of linear transformations that map the elements of a suitable vector space into each other, all of which would in turn usually be represented by a set of matrices like these:
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{| align="center" cellpadding="6" width="90%"
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<br>
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| align="center" |
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<pre>
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
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Table 2.  Matrix Representations of Permutations in Sym(3)
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|+ <math>\text{Matrix Representations of Permutations in}~ \operatorname{Sym}(3)</math>
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o---------o---------o---------o---------o---------o---------o
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|- style="background:#f0f0ff"
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|         |         |         |         |         |        |
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| width="16%" | <math>\operatorname{e}</math>
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|   e    |   f    |   g    |   h    |   i    |   j   |
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| width="16%" | <math>\operatorname{f}</math>
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|         |        |        |        |        |        |
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| width="16%" | <math>\operatorname{g}</math>
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o=========o=========o=========o=========o=========o=========o
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| width="16%" | <math>\operatorname{h}</math>
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|        |        |        |        |        |        |
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| width="16%" | <math>\operatorname{i}</math>
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|  1 0 0 |  0 0 1 | 0 1 0  |  1 0 0 |  0 0 1  | 0 1 0 |
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| width="16%" | <math>\operatorname{j}</math>
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|  0 1 0  |  1 0 0 | 0 0 1  |  0 0 1 |  0 1 0 |  1 0 0  |
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|-
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|  0 0 1 |  0 1 0 |  1 0 0 | 0 1 0 |  1 0 0 |  0 0 1 |
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|
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|        |        |        |        |        |        |
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<math>\begin{matrix}
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o---------o---------o---------o---------o---------o---------o
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1 & 0 & 0
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</pre>
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\\
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0 & 1 & 0
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\\
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0 & 0 & 1
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\end{matrix}</math>
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|
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<math>\begin{matrix}
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0 & 0 & 1
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\\
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1 & 0 & 0
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\\
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0 & 1 & 0
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\end{matrix}</math>
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|
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<math>\begin{matrix}
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0 & 1 & 0
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\\
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0 & 0 & 1
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\\
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1 & 0 & 0
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\end{matrix}</math>
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|
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<math>\begin{matrix}
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1 & 0 & 0
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\\
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0 & 0 & 1
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\\
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0 & 1 & 0
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\end{matrix}</math>
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|
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<math>\begin{matrix}
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0 & 0 & 1
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\\
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0 & 1 & 0
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\\
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1 & 0 & 0
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\end{matrix}</math>
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|
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<math>\begin{matrix}
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0 & 1 & 0
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\\
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1 & 0 & 0
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\\
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0 & 0 & 1
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\end{matrix}</math>
 
|}
 
|}
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<br>
    
The key to the mysteries of these matrices is revealed by noting that their coefficient entries are arrayed and overlaid on a place-mat marked like so:
 
The key to the mysteries of these matrices is revealed by noting that their coefficient entries are arrayed and overlaid on a place-mat marked like so:
Jon Awbrey
12,080

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