Linear dependence: Difference between revisions
Cmloegcmluin (talk | contribs) Created page with "Vector sets are '''collinear''', or '''linearly dependent''', when they share a common vector, meaning that they can form an identical vector through Wi..." |
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* To check if two mappings are collinear, 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 noncollinear, and otherwise the mappings are collinear and the result gives their shared vectors. | * To check if two mappings are collinear, 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 noncollinear, and otherwise the mappings are collinear and the result gives their shared vectors. | ||
* To check if two comma bases are collinear, we use a join. This process exactly parallels the process for checking two mappings for collinearity. 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 noncollinear, and otherwise they are collinear and the result gives their shared vectors. | * To check if two comma bases are collinear, we use a join. This process exactly parallels the process for checking two mappings for collinearity. 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 noncollinear, and otherwise they are collinear and the result gives their shared vectors. | ||
Certainly there are other ways to determine collinearity or not, but this method is handy because if they are collinear, then it also gives you the exact vectors by which they are collinear. | |||
=== Multivector collinearity === | === Multivector collinearity === | ||