Talk:Linear dependence: Difference between revisions

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:: - [[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 03:59, 19 December 2021 (UTC)

::: Alright, now that I understand the difference between collinearity and linear dependence clearly, I will remove the geometry section or drastically revise it. Thanks for explaining the difference; Dave (Keenan) and I were concerned about the difference and tried to figure it out for ourselves, but failed to do so.

::: Also, thanks for your advice to talk about rank deficiency instead. I agree 1-deficient could work well in some cases, such as the rows of a single temperament's mapping matrix. However, unfortunately, this is a bit different... it's about comparing two temperaments with each other. Let me give an example.

::: One comma basis for 11-limit meantone is {{bra|{{vector|4 -4 1 0 0}} {{vector|-1 2 0 -2 1}} {{vector|1 2 -3 1 0}}}}, and one basis for 11-limit mothra is {{bra|{{vector|4 -4 1 0 0}} {{vector|-1 2 0 -2 1}} {{vector|-7 -1 1 1 1}}}}. As you can see, they share 2 out of 3 vectors. So they only have 1 unshared vector.

::: I've been calling these types of temperament pairs "monononcollinear"; it's the compared pair that's monononcollinear, not either one of them individually. But that term is built on an incorrect definition of "collinear", so I should call them "mono- linearly independent", but that's no good. So as far as you know, there's no established term for this?

::: An example that is ''not'' monononcollinear: migration {{bra|{{vector|4 -4 1 0 0}} {{vector|1 2 -3 1 0}} {{vector|-1 5 0 0 -2}}}}, and baragon {{bra|{{vector|4 -4 1 0 0}} {{vector|-7 -1 1 1 1}} {{vector|7 -4 0 1 -1}}}}. They only share one vector, the meantone comma. This means that if you convert both of these matrices to multivectors, and sum them, the resultant multivector is nondecomposable, which means it doesn't represent a usable temperament. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 04:36, 19 December 2021 (UTC)

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 :: - [[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 03:59, 19 December 2021 (UTC)
+::: Alright, now that I understand the difference between collinearity and linear dependence clearly, I will remove the geometry section or drastically revise it. Thanks for explaining the difference; Dave (Keenan) and I were concerned about the difference and tried to figure it out for ourselves, but failed to do so.
+::: Also, thanks for your advice to talk about rank deficiency instead. I agree 1-deficient could work well in some cases, such as the rows of a single temperament's mapping matrix. However, unfortunately, this is a bit different... it's about comparing two temperaments with each other. Let me give an example.
+::: One comma basis for 11-limit meantone is {{bra|{{vector|4 -4 1 0 0}} {{vector|-1 2 0 -2 1}} {{vector|1 2 -3 1 0}}}}, and one basis for 11-limit mothra is {{bra|{{vector|4 -4 1 0 0}} {{vector|-1 2 0 -2 1}} {{vector|-7 -1 1 1 1}}}}. As you can see, they share 2 out of 3 vectors. So they only have 1 unshared vector.
+::: I've been calling these types of temperament pairs "monononcollinear"; it's the compared pair that's monononcollinear, not either one of them individually. But that term is built on an incorrect definition of "collinear", so I should call them "mono- linearly independent", but that's no good. So as far as you know, there's no established term for this?
+::: An example that is ''not'' monononcollinear: migration {{bra|{{vector|4 -4 1 0 0}} {{vector|1 2 -3 1 0}} {{vector|-1 5 0 0 -2}}}}, and baragon {{bra|{{vector|4 -4 1 0 0}} {{vector|-7 -1 1 1 1}} {{vector|7 -4 0 1 -1}}}}. They only share one vector, the meantone comma. This means that if you convert both of these matrices to multivectors, and sum them, the resultant multivector is nondecomposable, which means it doesn't represent a usable temperament. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 04:36, 19 December 2021 (UTC)