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 the operations [[meet and join]].

* 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 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 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 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.

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 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 the operations [[meet and join]].
-* 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 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 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 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.
 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>.