Treadstone 71
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"Suppose V is an n-dimensional vector space over an algebraically closed field F. Let T be a linear operator on V. Prove that there exists a cyclic vector for T <=> the minimal polynomial is equal to the characteristic polynomial of T."
(A cyclic vector is one such that (v,Tv,...,T^n-1 v) is a basis)
I got the => direction. I am having trouble with the backwards direction. Suppose the minimal polynomial and the characteristic polynomial of T are equal. Then the minimal polynomial has degree n, and since V is over an ac field, there are n roots, not necessarily distinct. But how do I produce a vector such that (v,Tv,...,T^(n-1)v) is linearly independent?
(A cyclic vector is one such that (v,Tv,...,T^n-1 v) is a basis)
I got the => direction. I am having trouble with the backwards direction. Suppose the minimal polynomial and the characteristic polynomial of T are equal. Then the minimal polynomial has degree n, and since V is over an ac field, there are n roots, not necessarily distinct. But how do I produce a vector such that (v,Tv,...,T^(n-1)v) is linearly independent?
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