Douglas Blumeyer's RTT How-To: Difference between revisions
Cmloegcmluin (talk | contribs) m →approximating JI: add link to search for yourself |
Cmloegcmluin (talk | contribs) m →null-space: more accurate terminology |
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We already have the tools to check that each of these commas’ vectors is tempered out individually by the map {{val|19 30 44}}; we learned this bit in the very first section: all we have to do is make sure that the comma maps to zero steps in this ET. But these matrices, the intersections of two of them at once, are another level of specialness. We can do even better with them. There’s actually a mathematical function which when input any one of these matrices will output {{monzo|{{val|19 30 44}}}}, thus demonstrating their equivalence with respect to it. | We already have the tools to check that each of these commas’ vectors is tempered out individually by the map {{val|19 30 44}}; we learned this bit in the very first section: all we have to do is make sure that the comma maps to zero steps in this ET. But these matrices, the intersections of two of them at once, are another level of specialness. We can do even better with them. There’s actually a mathematical function which when input any one of these matrices will output {{monzo|{{val|19 30 44}}}}, thus demonstrating their equivalence with respect to it. | ||
Remember that any one of these matrices is the null-space of {{monzo|{{val|19 30 44}}}}. So we are in effect undoing the effects of the null-space function. | Remember that any one of these matrices is the null-space of {{monzo|{{val|19 30 44}}}}. So we are in effect undoing the effects of the null-space function. Interestingly enough, as you'll soon see, it's about the same process to find the null-space as it is to undo it. Working this out by hand goes like this. | ||
First, transpose the matrix. That means the first column becomes the first row, the second column becomes the second row, etc. | First, transpose the matrix. That means the first column becomes the first row, the second column becomes the second row, etc. | ||
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</math> | </math> | ||
And ta-da! You’ve found the | And ta-da! You’ve found the mapping for which the comma basis we started is the null-space, and it is {{monzo|{{val|19 30 44}}}}. Feel free to try this with any other combination of two commas tempered out by this map. | ||
Now the null-space function, to take you from {{monzo|{{val|19 30 44}}}} back to the matrix, is pretty much the same thing, but a bit simpler. No need to transpose or reverse. Just start at the augmentation step: | Now the null-space function, to take you from {{monzo|{{val|19 30 44}}}} back to the matrix, is pretty much the same thing, but a bit simpler. No need to transpose or reverse. Just start at the augmentation step: | ||