Linear dependence: Difference between revisions

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==== For a given set of basis matrices, how to compute a basis for their linearly dependent vectors ====

A basis for the linearly dependent vectors of a set of basis matrices, or in other words, a linear-dependence basis <math>L_{\text{dep}}</math> can be computed using ~~the operations~~ [[~~meet and join~~]].

A basis for the linearly dependent vectors of a set of basis matrices, or in other words, a linear-dependence basis <math>L_{\text{dep}}</math> can be computed using [[temperament merging]].

* To check if two mappings are linearly dependent, we use a ~~meet~~. That is, we take the dual of each mapping to find its corresponding comma basis. Then we concatenate these two comma bases into one bigger comma basis. Finally, we take the dual of this comma basis to get back into mapping form. If this result is an empty matrix, then the mappings are linearly independent, and otherwise the mappings are linearly dependent and the result gives their linear-dependence basis.

* To check if two mappings are linearly dependent, we use a comma-merge. That is, we take the dual of each mapping to find its corresponding comma basis. Then we concatenate these two comma bases into one bigger comma basis. Finally, we take the dual of this comma basis to get back into mapping form. If this result is an empty matrix, then the mappings are linearly independent, and otherwise the mappings are linearly dependent and the result gives their linear-dependence basis.

* To check if two comma bases are linearly dependent, we use a ~~join~~. This process exactly parallels the process for checking two mappings for linear dependence. Take the duals of the comma bases to get two mappings, concatenate them into a single mapping, and take the dual again to get back to comma basis form. If the result is an empty matrix, the comma bases are linearly independent, and otherwise they are linearly dependent and the result gives a their linear-dependence basis.

* To check if two comma bases are linearly dependent, we use a map-merge. This process exactly parallels the process for checking two mappings for linear dependence. Take the duals of the comma bases to get two mappings, concatenate them into a single mapping, and take the dual again to get back to comma basis form. If the result is an empty matrix, the comma bases are linearly independent, and otherwise they are linearly dependent and the result gives a their linear-dependence basis.

Certainly there are other ways to determine linear dependency, but this method is handy because if the basis matrices ''are'' linearly dependent, then it also gives you <math>L_{\text{dep}}</math>.

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=== Wedge product ===

Linear dependence has an interesting effect on the wedge product, which otherwise produces the same result on a set of vectors as one would get by treating those same vectors as basis matrices and performing a ~~meet or join~~. The wedge product of any two linear dependent multivectors will have all zeros for entries, and thereby not represent an interesting new temperament (whereas the wedge product for linearly independent multivectors ''does'' represent an interesting new temperament sharing properties of the input temperaments) (and where the equivalent ~~meet or join~~ operation ~~from~~ linear algebra would provide such an interesting temperament). For more information, see: [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#Linearly dependent exception]]

Linear dependence has an interesting effect on the wedge product, which otherwise produces the same result on a set of vectors as one would get by treating those same vectors as basis matrices and performing a temperament merge. The wedge product of any two linear dependent multivectors will have all zeros for entries, and thereby not represent an interesting new temperament (whereas the wedge product for linearly independent multivectors ''does'' represent an interesting new temperament sharing properties of the input temperaments) (and where the equivalent temperament merge operation in linear algebra would provide such an interesting temperament). For more information, see: [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#Linearly dependent exception]]

=== Temperament arithmetic ===

@@ Line 27: / Line 27: @@
 ==== For a given set of basis matrices, how to compute a basis for their linearly dependent vectors ====
-A basis for the linearly dependent vectors of a set of basis matrices, or in other words, a linear-dependence basis <math>L_{\text{dep}}</math> can be computed using the operations [[meet and join]].
+A basis for the linearly dependent vectors of a set of basis matrices, or in other words, a linear-dependence basis <math>L_{\text{dep}}</math> can be computed using [[temperament merging]].
-* To check if two mappings are linearly dependent, we use a meet. That is, we take the dual of each mapping to find its corresponding comma basis. Then we concatenate these two comma bases into one bigger comma basis. Finally, we take the dual of this comma basis to get back into mapping form. If this result is an empty matrix, then the mappings are linearly independent, and otherwise the mappings are linearly dependent and the result gives their linear-dependence basis.
+* To check if two mappings are linearly dependent, we use a comma-merge. That is, we take the dual of each mapping to find its corresponding comma basis. Then we concatenate these two comma bases into one bigger comma basis. Finally, we take the dual of this comma basis to get back into mapping form. If this result is an empty matrix, then the mappings are linearly independent, and otherwise the mappings are linearly dependent and the result gives their linear-dependence basis.
-* To check if two comma bases are linearly dependent, we use a join. This process exactly parallels the process for checking two mappings for linear dependence. Take the duals of the comma bases to get two mappings, concatenate them into a single mapping, and take the dual again to get back to comma basis form. If the result is an empty matrix, the comma bases are linearly independent, and otherwise they are linearly dependent and the result gives a their linear-dependence basis.
+* To check if two comma bases are linearly dependent, we use a map-merge. This process exactly parallels the process for checking two mappings for linear dependence. Take the duals of the comma bases to get two mappings, concatenate them into a single mapping, and take the dual again to get back to comma basis form. If the result is an empty matrix, the comma bases are linearly independent, and otherwise they are linearly dependent and the result gives a their linear-dependence basis.
 Certainly there are other ways to determine linear dependency, but this method is handy because if the basis matrices ''are'' linearly dependent, then it also gives you <math>L_{\text{dep}}</math>.
@@ Line 53: / Line 53: @@
 === Wedge product ===
-Linear dependence has an interesting effect on the wedge product, which otherwise produces the same result on a set of vectors as one would get by treating those same vectors as basis matrices and performing a meet or join. The wedge product of any two linear dependent multivectors will have all zeros for entries, and thereby not represent an interesting new temperament (whereas the wedge product for linearly independent multivectors ''does'' represent an interesting new temperament sharing properties of the input temperaments) (and where the equivalent meet or join operation from linear algebra would provide such an interesting temperament). For more information, see: [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#Linearly dependent exception]]
+Linear dependence has an interesting effect on the wedge product, which otherwise produces the same result on a set of vectors as one would get by treating those same vectors as basis matrices and performing a temperament merge. The wedge product of any two linear dependent multivectors will have all zeros for entries, and thereby not represent an interesting new temperament (whereas the wedge product for linearly independent multivectors ''does'' represent an interesting new temperament sharing properties of the input temperaments) (and where the equivalent temperament merge operation in linear algebra would provide such an interesting temperament). For more information, see: [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#Linearly dependent exception]]
 === Temperament arithmetic ===