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

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[[File:Different nestings.png|400px|thumb|left|'''Figure 5a.''' How to write matrices in terms of either columns/vectors/commas or rows/covectors/maps.]]

We can extend our angle bracket notation (technically called [https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation bra-ket notation, or Dirac notation]<ref>~~Dirac~~ notation comes to RTT from quantum mechanics, not algebra.</ref>) to handle matrices by nesting rows inside columns, or columns inside rows ''(see Figure 5a)''. For example, we could have written our comma basis like this: {{map|{{vector|-4 4 -1}} {{vector|-10 -1 5}}}}. Starting from the outside, the {{map|}} tells us to think in terms of a row. It's just that this covector isn't a covector of numbers, like the ones we've gotten used to by now, but rather a covector of ''columns of'' numbers. So this row houses two such columns. Alternatively, we could have written this same matrix like {{vector|{{map|-4 -10}} {{map|4 -1}} {{map|-1 5}}}}, but that would obscure the fact that it is the combination of two familiar commas (but that notation ''would'' be useful for expressing a matrix built out of multiple maps, as we will soon see).

We can extend our angle bracket notation (technically called [https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation bra-ket notation, or Dirac notation]<ref>Bra-ket notation comes to RTT from quantum mechanics, not algebra.</ref>) to handle matrices by nesting rows inside columns, or columns inside rows ''(see Figure 5a)''. For example, we could have written our comma basis like this: {{map|{{vector|-4 4 -1}} {{vector|-10 -1 5}}}}. Starting from the outside, the {{map|}} tells us to think in terms of a row. It's just that this covector isn't a covector of numbers, like the ones we've gotten used to by now, but rather a covector of ''columns of'' numbers. So this row houses two such columns. Alternatively, we could have written this same matrix like {{vector|{{map|-4 -10}} {{map|4 -1}} {{map|-1 5}}}}, but that would obscure the fact that it is the combination of two familiar commas (but that notation ''would'' be useful for expressing a matrix built out of multiple maps, as we will soon see).

Sometimes a comma basis may have only a single comma. That’s okay. A single vector can become a matrix. To disambiguate this situation, you could put the vector inside a covector, like this: {{map|{{vector|-4 4 -1}}}}. Similarly, a single covector can become a matrix, by nesting inside a vector, like this: {{vector|{{map|19 30 44}}}}.

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|(column) vector, matrix column

|-

!multilinear algebra structure<ref>RTT uses multilinear algebra structures, but does not use the conventional tensor (multi-dimensional array) representations of them; instead, it uses two copies of exterior algebra, one on each side of the duality, and an extended version of ~~Dirac~~ notation.</ref>

!multilinear algebra structure<ref>RTT uses multilinear algebra structures, but does not use the conventional tensor (multi-dimensional array) representations of them; instead, it uses two copies of exterior algebra, one on each side of the duality, and an extended version of bra-ket notation.</ref>

|covector

|list of covectors

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|vector

|-

!extended ~~Dirac~~ notation representation

!extended bra-ket notation representation

|bra

|ket of bras

@@ Line 479: / Line 479: @@
 [[File:Different nestings.png|400px|thumb|left|'''Figure 5a.''' How to write matrices in terms of either columns/vectors/commas or rows/covectors/maps.]]
-We can extend our angle bracket notation (technically called [https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation bra-ket notation, or Dirac notation]<ref>Dirac notation comes to RTT from quantum mechanics, not algebra.</ref>) to handle matrices by nesting rows inside columns, or columns inside rows ''(see Figure 5a)''. For example, we could have written our comma basis like this: {{map|{{vector|-4 4 -1}} {{vector|-10 -1 5}}}}. Starting from the outside, the {{map|}} tells us to think in terms of a row. It's just that this covector isn't a covector of numbers, like the ones we've gotten used to by now, but rather a covector of ''columns of'' numbers. So this row houses two such columns. Alternatively, we could have written this same matrix like {{vector|{{map|-4 -10}} {{map|4 -1}} {{map|-1 5}}}}, but that would obscure the fact that it is the combination of two familiar commas (but that notation ''would'' be useful for expressing a matrix built out of multiple maps, as we will soon see).
+We can extend our angle bracket notation (technically called [https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation bra-ket notation, or Dirac notation]<ref>Bra-ket notation comes to RTT from quantum mechanics, not algebra.</ref>) to handle matrices by nesting rows inside columns, or columns inside rows ''(see Figure 5a)''. For example, we could have written our comma basis like this: {{map|{{vector|-4 4 -1}} {{vector|-10 -1 5}}}}. Starting from the outside, the {{map|}} tells us to think in terms of a row. It's just that this covector isn't a covector of numbers, like the ones we've gotten used to by now, but rather a covector of ''columns of'' numbers. So this row houses two such columns. Alternatively, we could have written this same matrix like {{vector|{{map|-4 -10}} {{map|4 -1}} {{map|-1 5}}}}, but that would obscure the fact that it is the combination of two familiar commas (but that notation ''would'' be useful for expressing a matrix built out of multiple maps, as we will soon see).
 Sometimes a comma basis may have only a single comma. That’s okay. A single vector can become a matrix. To disambiguate this situation, you could put the vector inside a covector, like this: {{map|{{vector|-4 4 -1}}}}. Similarly, a single covector can become a matrix, by nesting inside a vector, like this: {{vector|{{map|19 30 44}}}}.
@@ Line 1,460: / Line 1,460: @@
 |(column) vector, matrix column
 |-
-!multilinear algebra structure<ref>RTT uses multilinear algebra structures, but does not use the conventional tensor (multi-dimensional array) representations of them; instead, it uses two copies of exterior algebra, one on each side of the duality, and an extended version of Dirac notation.</ref>
+!multilinear algebra structure<ref>RTT uses multilinear algebra structures, but does not use the conventional tensor (multi-dimensional array) representations of them; instead, it uses two copies of exterior algebra, one on each side of the duality, and an extended version of bra-ket notation.</ref>
 |covector
 |list of covectors
@@ Line 1,468: / Line 1,468: @@
 |vector
 |-
-!extended Dirac notation representation
+!extended bra-ket notation representation
 |bra
 |ket of bras