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]) 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: {{val|{{monzo|-4 4 -1}} {{monzo|-10 -1 5}}}}. Starting from the outside, the {{val|}} 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 {{monzo|{{val|-4 -10}} {{val|4 -1}} {{val|-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>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: {{val|{{monzo|-4 4 -1}} {{monzo|-10 -1 5}}}}. Starting from the outside, the {{val|}} 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 {{monzo|{{val|-4 -10}} {{val|4 -1}} {{val|-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: {{val|{{monzo|-4 4 -1}}}}. Similarly, a single covector can become a matrix, by nesting inside a vector, like this: {{monzo|{{val|19 30 44}}}}.

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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]) 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: {{val|{{monzo|-4 4 -1}} {{monzo|-10 -1 5}}}}. Starting from the outside, the {{val|}} 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 {{monzo|{{val|-4 -10}} {{val|4 -1}} {{val|-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>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: {{val|{{monzo|-4 4 -1}} {{monzo|-10 -1 5}}}}. Starting from the outside, the {{val|}} 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 {{monzo|{{val|-4 -10}} {{val|4 -1}} {{val|-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: {{val|{{monzo|-4 4 -1}}}}. Similarly, a single covector can become a matrix, by nesting inside a vector, like this: {{monzo|{{val|19 30 44}}}}.