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

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[[File:Different nestings.png|400px|thumb|left|'''Figure 4a.''' How to write matrices in terms of either columns/vectors/commas or rows/covectors/maps.]]

We can extend our angle bracket notation (technically called ~~braket~~ or Dirac notation) to handle matrices by nesting rows inside columns, or columns inside rows ''(see Figure 4a)''. 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. So this row houses two 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 or Dirac notation]) to handle matrices by nesting rows inside columns, or columns inside rows ''(see Figure 4a)''. 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 4a.''' How to write matrices in terms of either columns/vectors/commas or rows/covectors/maps.]]
-We can extend our angle bracket notation (technically called braket or Dirac notation) to handle matrices by nesting rows inside columns, or columns inside rows ''(see Figure 4a)''. 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. So this row houses two 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 or Dirac notation]) to handle matrices by nesting rows inside columns, or columns inside rows ''(see Figure 4a)''. 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}}}}.