Domain basis: Difference between revisions
Cmloegcmluin (talk | contribs) →Intersecting: applications |
Cmloegcmluin (talk | contribs) →Intersecting: attempt to explain how it works |
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# The results we want are in the bottom-right. We take only half-columns, from the bottom half, and only half-columns where their corresponding top-half are all zeros (which will only happen to columns that are sorted to the right side). | # The results we want are in the bottom-right. We take only half-columns, from the bottom half, and only half-columns where their corresponding top-half are all zeros (which will only happen to columns that are sorted to the right side). | ||
# Canonicalize. | # Canonicalize. | ||
The reason this works is that wherever the corresponding top-half columns are all zeros, this was achieved through linear combinations of vectors from both interval bases, which means the information below them represents vectors that are in both of them. In other words, if <math>(x, x) + (y, 0) = (0, z)</math> and <math>x</math> is in <math>B_1</math> and <math>y</math> is in <math>B_2</math>, then we must have <math>x + y = 0</math> and <math>z = x</math><ref>credit this explanation to Tom Price on Discord</ref>. We're sort of abusing HNF as a way to solve a system, kind of like [[Douglas Blumeyer's RTT How-To#Null-space|when we calculate the null-space]]<ref>Credit this explanation to Sintel on Discord</ref>. | |||
== But first: a gentle introduction == | == But first: a gentle introduction == | ||