Assume

show that:

If and

then

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- Dec 13th 2009, 12:36 AMXingyuanProblem of rank
Assume

show that:

If and

then - Dec 13th 2009, 12:27 PMNonCommAlg
let and be the corresponding linear transformations defined by let's put and

since we have thus, using the rank nullity theore, we have and hence

on the other hand, since and we have, again by the rank-nullity theorem, now suppose

then and thus i.e. hence and therefore so

finally, using (1) and (2) and the rank-nullity theorem, completing the proof is easy: