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

Line 607:

And ta-da! You’ve found the mapping for which the comma basis we started with is a basis for the null-space, and it is {{vector|{{map|19 30 44}}}}. Feel free to try this with any other combination of two commas tempered out by this map.

So why why did we need to do those extra reversals at the beginning and end? Besides, I never said we ''must'' find get the zeroes in the top half on the top right of the augmented matrix when doing column Gaussian elimination, so wasn't rearranging the columns pointless? Well, the reason I told you to do it was because if you're going to adapt this process to a math program like Wolfram Alpha or perhaps even general computer code, you ''will'' need that step, because the way the null-space algorithm is implemented, it ''will'' try to get those zeroes on the right. So I understand this is a bit of a hand-wavy answer, and perhaps one day someone else can edit this with harder facts. But based on observation, if you do not do the reversing, you end up with an incorrect answer, and in particular, it has got zeroes on the wrong side of the matrix than you would expect. Unfortunately, to the best of my Wolfram Alpha ability, I'm unable to make the entire process work in one go (it is possible with Wolfram Language in general, which you can play with in a Wolfram computable notebook), so here is just the part where you take the null-space of the already transposed and reversed comma basis:

{| class="wikitable"

Line 617:

Line 619:

|}

Now the null-space function, to take you from {{vector|{{map|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 {{vector|{{map|19 30 44}}}} back to the matrix, is pretty much the same thing, but simpler! No need to transpose or reverse like that. Just start at the augmentation step:

<math>

Line 681:

Line 683:

</math>

So that’s not any of the commas we’ve looked at so far (it’s the [[19-comma]] and the [[acute limma]]). But it is clear to see that either of them would be tempered out by 19-ET (no need to map by hand — just look at these commas side-by-side with the map {{vector|{{map|19 30 44}}}} and it should be apparent).

So that’s not any of the commas we’ve looked at so far (it’s the [[19-comma]] and the [[acute limma]]). But it is clear to see that either of them would be tempered out by 19-ET (no need to map by hand — just look at these commas side-by-side with the map {{vector|{{map|19 30 44}}}} and it should be apparent). We're done!

{| class="wikitable"

@@ Line 607: / Line 607: @@
 And ta-da! You’ve found the mapping for which the comma basis we started with is a basis for the null-space, and it is {{vector|{{map|19 30 44}}}}. Feel free to try this with any other combination of two commas tempered out by this map.
+So why why did we need to do those extra reversals at the beginning and end? Besides, I never said we ''must'' find get the zeroes in the top half on the top right of the augmented matrix when doing column Gaussian elimination, so wasn't rearranging the columns pointless? Well, the reason I told you to do it was because if you're going to adapt this process to a math program like Wolfram Alpha or perhaps even general computer code, you ''will'' need that step, because the way the null-space algorithm is implemented, it ''will'' try to get those zeroes on the right. So I understand this is a bit of a hand-wavy answer, and perhaps one day someone else can edit this with harder facts. But based on observation, if you do not do the reversing, you end up with an incorrect answer, and in particular, it has got zeroes on the wrong side of the matrix than you would expect. Unfortunately, to the best of my Wolfram Alpha ability, I'm unable to make the entire process work in one go (it is possible with Wolfram Language in general, which you can play with in a Wolfram computable notebook), so here is just the part where you take the null-space of the already transposed and reversed comma basis:
 {| class="wikitable"
@@ Line 617: / Line 619: @@
 |}
-Now the null-space function, to take you from {{vector|{{map|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 {{vector|{{map|19 30 44}}}} back to the matrix, is pretty much the same thing, but simpler! No need to transpose or reverse like that. Just start at the augmentation step:
 <math>
@@ Line 681: / Line 683: @@
 </math>
-So that’s not any of the commas we’ve looked at so far (it’s the [[19-comma]] and the [[acute limma]]). But it is clear to see that either of them would be tempered out by 19-ET (no need to map by hand — just look at these commas side-by-side with the map {{vector|{{map|19 30 44}}}} and it should be apparent).
+So that’s not any of the commas we’ve looked at so far (it’s the [[19-comma]] and the [[acute limma]]). But it is clear to see that either of them would be tempered out by 19-ET (no need to map by hand — just look at these commas side-by-side with the map {{vector|{{map|19 30 44}}}} and it should be apparent). We're done!
 {| class="wikitable"