Dual list: Difference between revisions

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This page gives a more mathematical take on this topic. For a basic introduction, see [[comma basis]].

: ''This page gives a more mathematical take on this topic. For a basic introduction, see [[comma basis]].''

== Definition ==

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== Examples ==

dl[{{map| 31 49 72 87 107 }}, {{map| 41 65 95 115 142 }}, {{map| 72 114 167 202 249 }}]) {{nowrap|{{=}} ~~{{!(}}~~{{vector| -3 2 -1 2 -1 }}, {{vector| -2 -3 -1 2 1 }}, {{vector| -2 0 3 -3 1 }}~~{{)!}}~~}}

dl[{{map| 31 49 72 87 107 }}, {{map| 41 65 95 115 142 }}, {{map| 72 114 167 202 249 }}]) {{nowrap|{{=}} [{{vector| -3 2 -1 2 -1 }}, {{vector| -2 -3 -1 2 1 }}, {{vector| -2 0 3 -3 1 }}]}}

In this first example, the mapping contains a redundant row; the third row is a linear combination of the other two ({{nowrap|31 + 41 {{=}} 72|49 + 65 {{=}} 114}}, etc). Therefore, the dual list operation returns a comma basis with nullity 3, because the rank is truly 2 (and the rank and nullity must sum to the dimensionality, which is 5 here, because there are 5 terms in each map and comma, i.e. it is 11-limit).

dl[{{vector| -3 2 -1 2 -1 }}, {{vector| -2 -3 -1 2 1 }}, {{vector| -2 0 3 -3 1 }}]) {{nowrap|{{=}} ~~{{!(}}~~{{map| 1 1 3 3 2 }}, {{map| 0 6 -7 -2 15 }}~~{{)!}}~~}}

dl[{{vector| -3 2 -1 2 -1 }}, {{vector| -2 -3 -1 2 1 }}, {{vector| -2 0 3 -3 1 }}]) {{nowrap|{{=}} [{{map| 1 1 3 3 2 }}, {{map| 0 6 -7 -2 15 }}]}}

dl[{{map| 1 1 3 3 2 }}, {{map| 0 6 -7 -2 15 }}]) {{nowrap|{{=}} ~~{{!(}}~~{{vector| -3 2 -1 2 -1 }}, {{vector| -2 -3 -1 2 1 }}, {{vector| -2 0 3 -3 1 }}~~{{)!}}~~}}

dl[{{map| 1 1 3 3 2 }}, {{map| 0 6 -7 -2 15 }}]) {{nowrap|{{=}} [{{vector| -3 2 -1 2 -1 }}, {{vector| -2 -3 -1 2 1 }}, {{vector| -2 0 3 -3 1 }}]}}

[[Category:Regular temperament theory]]

[[Category:Math]]

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-{{Legacy}}
+{{inacc}}
-This page gives a more mathematical take on this topic. For a basic introduction, see [[comma basis]].
+{{todo|intro|inline=1}}
+: ''This page gives a more mathematical take on this topic. For a basic introduction, see [[comma basis]].''
 == Definition ==
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 == Examples ==
-dl[{{map| 31 49 72 87 107 }}, {{map| 41 65 95 115 142 }}, {{map| 72 114 167 202 249 }}]) {{nowrap|{{=}} {{!(}}{{vector| -3 2 -1 2 -1 }}, {{vector| -2 -3 -1 2 1 }}, {{vector| -2 0 3 -3 1 }}{{)!}}}}
+dl[{{map| 31 49 72 87 107 }}, {{map| 41 65 95 115 142 }}, {{map| 72 114 167 202 249 }}]) {{nowrap|{{=}} [{{vector| -3 2 -1 2 -1 }}, {{vector| -2 -3 -1 2 1 }}, {{vector| -2 0 3 -3 1 }}]}}
 In this first example, the mapping contains a redundant row; the third row is a linear combination of the other two ({{nowrap|31 + 41 {{=}} 72|49 + 65 {{=}} 114}}, etc). Therefore, the dual list operation returns a comma basis with nullity 3, because the rank is truly 2 (and the rank and nullity must sum to the dimensionality, which is 5 here, because there are 5 terms in each map and comma, i.e. it is 11-limit).
-dl[{{vector| -3 2 -1 2 -1 }}, {{vector| -2 -3 -1 2 1 }}, {{vector| -2 0 3 -3 1 }}]) {{nowrap|{{=}} {{!(}}{{map| 1 1 3 3 2 }}, {{map| 0 6 -7 -2 15 }}{{)!}}}}
+dl[{{vector| -3 2 -1 2 -1 }}, {{vector| -2 -3 -1 2 1 }}, {{vector| -2 0 3 -3 1 }}]) {{nowrap|{{=}} [{{map| 1 1 3 3 2 }}, {{map| 0 6 -7 -2 15 }}]}}
-dl[{{map| 1 1 3 3 2 }}, {{map| 0 6 -7 -2 15 }}]) {{nowrap|{{=}} {{!(}}{{vector| -3 2 -1 2 -1 }}, {{vector| -2 -3 -1 2 1 }}, {{vector| -2 0 3 -3 1 }}{{)!}}}}
+dl[{{map| 1 1 3 3 2 }}, {{map| 0 6 -7 -2 15 }}]) {{nowrap|{{=}} [{{vector| -3 2 -1 2 -1 }}, {{vector| -2 -3 -1 2 1 }}, {{vector| -2 0 3 -3 1 }}]}}
 [[Category:Regular temperament theory]]
 [[Category:Math]]