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**realpart1/2** f:V-->V is an idempotent (i.e. f(f(v))=f(v)) linear map of an n-dimensional F-vector space V. I need to prove that there is a basis {v1,v2,...,vn} and a natural number r: 0<=r<=n so that f(vi)=vi for 0<=i<=r and f(vi)=0 for r+1<=i<=n.

I think it has something to do with an intersection of N(f) and V having to have something other than 0, but I'm still struggling with the actual proof. I think it has something to do with the idempotency of f.

An easy to follow formalized proof would be greatly appreciated.