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

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# 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 map 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><ref>You may also sometimes see "Hodge dual", or "Hodge star". This operation is not another name for the dual operation. It is a linear algebra operation which works as a limited substitute for the exterior algebra complement operation which RTT's dual is based on. The limitation is that it only works when the rank is 2. This is because when rank is 2, bi(co)vectors can be represented as skew-symmetric matrices (see: https://en.wikipedia.org/wiki/Bivector#Matrices), which gives you access to some extra linear algebra utilities such as this.</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.</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.]]

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 # 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 map 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><ref>You may also sometimes see "Hodge dual", or "Hodge star". This operation is not another name for the dual operation. It is a linear algebra operation which works as a limited substitute for the exterior algebra complement operation which RTT's dual is based on. The limitation is that it only works when the rank is 2. This is because when rank is 2, bi(co)vectors can be represented as skew-symmetric matrices (see: https://en.wikipedia.org/wiki/Bivector#Matrices), which gives you access to some extra linear algebra utilities such as this.</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.</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.]]