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

Line 1,328:

# Just as a vector is the dual of a covector, we also have a '''multivector''' which is the dual of a multicovector. Analogously, we call the thing the multivector represents a '''multicomma'''.

# We can calculate a multicomma from a comma basis much in the same way we can calculate a multimap from a mapping-row basis

# We can convert between multimaps and multicommas using an operation called “taking '''[[the dual]]'''”<ref>This operation uses the same process as is used for finding the complement in exterior algebra, however, whereas exterior algebra does not convert between vectors and covectors (it can be used on either one, staying within that category), with RTT's dual you switch which type it is as the last step.</ref>, which basically involves reversing the order of terms and changing the signs of some of them.

# We can convert between multimaps and multicommas using an operation called “taking '''[[the dual]]'''”<ref>This operation uses the same process as is used for finding the complement in exterior algebra, however, whereas exterior algebra does not convert between vectors and covectors (it can be used on either one, staying within that category), with RTT's dual you switch which type it is as the last step.

{| class="wikitable"

|+'''Table 6(footnote).''' Three similar duals

!<math>d</math>

!<math>r</math>

!<math>d - r</math>

!<math>{d \choose r}</math>

!<math>{d \choose {d - r}}</math>

!count

|-

|3

|2

|1

|<math>(2,3) (2,5) (3,5)</math>

|<math>(2) (3) (5)</math>

|3

|-

|4

|3

|1

|<math>(2,3,5) (2,3,7) (2,5,7) (3,5,7)</math>

|<math>(2) (3) (5) (7)</math>

|4

|-

|5

|3

|2

|<math>(2,3,5) (2,3,7) (2,3,11) (2,5,7) (2,5,11) (2,7,11) (3,5,7) (3,5,11) (3,7,11) (5,7,11)</math>

|<math>(2,3) (2,5) (2,7) (2,11) (3,5), (3,7) (3,11) (5,7) (5,11) (7,11)</math>

|10

|}

</ref>, which basically involves reversing the order of terms and changing the signs of some of them.

[[File:Algebra notation.png|300px|thumb|right|'''Figure 6a.''' RTT bracket notation comparison.]]

@@ Line 1,328: / Line 1,328: @@
 # Just as a vector is the dual of a covector, we also have a '''multivector''' which is the dual of a multicovector. Analogously, we call the thing the multivector represents a '''multicomma'''.
 # We can calculate a multicomma from a comma basis much in the same way we can calculate a multimap from a mapping-row basis
-# We can convert between multimaps and multicommas using an operation called “taking '''[[the dual]]'''”<ref>This operation uses the same process as is used for finding the complement in exterior algebra, however, whereas exterior algebra does not convert between vectors and covectors (it can be used on either one, staying within that category), with RTT's dual you switch which type it is as the last step.</ref>, which basically involves reversing the order of terms and changing the signs of some of them.
+# We can convert between multimaps and multicommas using an operation called “taking '''[[the dual]]'''”<ref>This operation uses the same process as is used for finding the complement in exterior algebra, however, whereas exterior algebra does not convert between vectors and covectors (it can be used on either one, staying within that category), with RTT's dual you switch which type it is as the last step.
+{| class="wikitable"
+|+'''Table 6(footnote).''' Three similar duals
+!<math>d</math>
+!<math>r</math>
+!<math>d - r</math>
+!<math>{d \choose r}</math>
+!<math>{d \choose {d - r}}</math>
+!count
+|-
+|3
+|2
+|1
+|<math>(2,3) (2,5) (3,5)</math>
+|<math>(2) (3) (5)</math>
+|3
+|-
+|4
+|3
+|1
+|<math>(2,3,5) (2,3,7) (2,5,7) (3,5,7)</math>
+|<math>(2) (3) (5) (7)</math>
+|4
+|-
+|5
+|3
+|2
+|<math>(2,3,5) (2,3,7) (2,3,11) (2,5,7) (2,5,11) (2,7,11) (3,5,7) (3,5,11) (3,7,11) (5,7,11)</math>
+|<math>(2,3) (2,5) (2,7) (2,11) (3,5), (3,7) (3,11) (5,7) (5,11) (7,11)</math>
+|10
+|}
+</ref>, which basically involves reversing the order of terms and changing the signs of some of them.
 [[File:Algebra notation.png|300px|thumb|right|'''Figure 6a.''' RTT bracket notation comparison.]]