Douglas Blumeyer's RTT How-To: Difference between revisions

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If a comma basis is the name for the matrix made out of commas, then we could say a “'''mapping'''” is the name for the matrix made out of maps. Why isn't this one a "basis", you ask? Well, it can be thought of as a basis too. It depends on the context. When you use the word "mapping" for it, you're treating it like a function, or a machine: it takes in intervals, and spits out new forms of intervals. That's how we've been using it here. But in other places, you may be thinking of this matrix as a basis for the infinite space of possible maps that could be combined to produce a matrix which works the same way as a given mapping, i.e. it tempers out the same commas. In these contexts, it might make more sense to call such a mapping matrix a "mapping-row basis".

And now you wonder why it's not just "~~mapping-row~~ basis". Well, that's answerable too. It's because "map" is the analogous term to an "interval", but we're looking for the analogous term to a "comma". A comma is an interval which is tempered out. So we need a word that means a map which tempers out, and that term is "mapping-row".

And now you wonder why it's not just "map basis". Well, that's answerable too. It's because "map" is the analogous term to an "interval", but we're looking for the analogous term to a "comma". A comma is an interval which is tempered out. So we need a word that means a map which tempers out, and that term is "mapping-row".

So, yes, that's right: maps are similar to commas insofar as — once you have more than one of them in your matrix — the possibilities for individual members immediately go infinite. Technically speaking, though, while a comma basis is a basis of the null-space of the mapping, a mapping-row basis is a ''row''-basis of the ''row''-space of the mapping.

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 If a comma basis is the name for the matrix made out of commas, then we could say a “'''mapping'''” is the name for the matrix made out of maps. Why isn't this one a "basis", you ask? Well, it can be thought of as a basis too. It depends on the context. When you use the word "mapping" for it, you're treating it like a function, or a machine: it takes in intervals, and spits out new forms of intervals. That's how we've been using it here. But in other places, you may be thinking of this matrix as a basis for the infinite space of possible maps that could be combined to produce a matrix which works the same way as a given mapping, i.e. it tempers out the same commas. In these contexts, it might make more sense to call such a mapping matrix a "mapping-row basis".
-And now you wonder why it's not just "mapping-row basis". Well, that's answerable too. It's because "map" is the analogous term to an "interval", but we're looking for the analogous term to a "comma". A comma is an interval which is tempered out. So we need a word that means a map which tempers out, and that term is "mapping-row".
+And now you wonder why it's not just "map basis". Well, that's answerable too. It's because "map" is the analogous term to an "interval", but we're looking for the analogous term to a "comma". A comma is an interval which is tempered out. So we need a word that means a map which tempers out, and that term is "mapping-row".
 So, yes, that's right: maps are similar to commas insofar as — once you have more than one of them in your matrix — the possibilities for individual members immediately go infinite. Technically speaking, though, while a comma basis is a basis of the null-space of the mapping, a mapping-row basis is a ''row''-basis of the ''row''-space of the mapping.