- #1
swampwiz
- 571
- 83
Specifically, an elemental shear operation.
I can prove how any existing linear dependency that exists persists with the new sheared vector, but I can't seem to prove that this shear operation does not introduce a linear dependency. I can see how the vector space between the 2 rows involved does not change since the situation of those vectors being non-parallel has the updated row still not being parallel to the other, and for the situation in which they are parallel, the updated row remains parallel. My goal is to be able to prove that the rank does not change in a shear operation.
EDIT: I think I can justify the contention that there is no new linear dependency since I do not introduce an equation of the form of the zero vector (or any constant vector) to a coefficient sum of row vectors - but this seems weak.
I can prove how any existing linear dependency that exists persists with the new sheared vector, but I can't seem to prove that this shear operation does not introduce a linear dependency. I can see how the vector space between the 2 rows involved does not change since the situation of those vectors being non-parallel has the updated row still not being parallel to the other, and for the situation in which they are parallel, the updated row remains parallel. My goal is to be able to prove that the rank does not change in a shear operation.
EDIT: I think I can justify the contention that there is no new linear dependency since I do not introduce an equation of the form of the zero vector (or any constant vector) to a coefficient sum of row vectors - but this seems weak.
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