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 matrix much in the same way we can calculate a multimap from a mapping matrix

# We can convert between multimaps and multicommas using an operation called “taking the '''complement'''”<ref>Elsewhere on the wiki you may find the complement operation called "taking [[the dual]]", or even the dual of a multimap being called simply "the dual". ~~An alternate name for "taking the complement" may be "taking the Hodge dual", but in~~ these materials, I am using the dual to refer to the general case, while the specific case of the dual of a multimap is a multicomma and the operation to get from one of these to its dual is called taking the complement (whereas to get to the dual of a mapping, which is a comma basis, the operation is called taking the null-space).</ref>, which basically involves reversing the order of terms and negating some of them.

# We can convert between multimaps and multicommas using an operation called “taking the '''complement'''”<ref>Elsewhere on the wiki you may find the complement operation called "taking [[the dual]]", or even the dual of a multimap being called simply "the dual". In these materials, I am using the dual to refer to the general case, while the specific case of the dual of a multimap is a multicomma and the operation to get from one of these to its dual is called taking the complement (whereas to get to the dual of a mapping, which is a comma basis, the operation is called taking the null-space).</ref><ref>You may also sometimes see "Hodge dual" used where you'd expect to see the complement operation. The Hodge star operation, or Hodge dual operation, is not another name for the complement operation. It is a linear algebra operation which works as a limited substitute for the exterior algebra operation. The limitation is that it only works when the rank is 2. This is because when rank is 2, bicovectors 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 Hodge star.</ref>, which basically involves reversing the order of terms and negating 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 matrix much in the same way we can calculate a multimap from a mapping matrix
-# We can convert between multimaps and multicommas using an operation called “taking the '''complement'''”<ref>Elsewhere on the wiki you may find the complement operation called "taking [[the dual]]", or even the dual of a multimap being called simply "the dual". An alternate name for "taking the complement" may be "taking the Hodge dual", but in these materials, I am using the dual to refer to the general case, while the specific case of the dual of a multimap is a multicomma and the operation to get from one of these to its dual is called taking the complement (whereas to get to the dual of a mapping, which is a comma basis, the operation is called taking the null-space).</ref>, which basically involves reversing the order of terms and negating some of them.
+# We can convert between multimaps and multicommas using an operation called “taking the '''complement'''”<ref>Elsewhere on the wiki you may find the complement operation called "taking [[the dual]]", or even the dual of a multimap being called simply "the dual". In these materials, I am using the dual to refer to the general case, while the specific case of the dual of a multimap is a multicomma and the operation to get from one of these to its dual is called taking the complement (whereas to get to the dual of a mapping, which is a comma basis, the operation is called taking the null-space).</ref><ref>You may also sometimes see "Hodge dual" used where you'd expect to see the complement operation. The Hodge star operation, or Hodge dual operation, is not another name for the complement operation. It is a linear algebra operation which works as a limited substitute for the exterior algebra operation. The limitation is that it only works when the rank is 2. This is because when rank is 2, bicovectors 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 Hodge star.</ref>, which basically involves reversing the order of terms and negating some of them.
 [[File:Algebra notation.png|300px|thumb|right|'''Figure 6a.''' RTT bracket notation comparison.]]