Let T be a nilpotent operator on an n-dimensional vector space V, and suppose that p is the smallest positive integer fo

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Let T Be A Nilpotent Operator On An N Dimensional Vector Space V And Suppose That P Is The Smallest Positive Integer Fo 1 (30.2 KiB) Viewed 26 times

Let T be a nilpotent operator on an n-dimensional vector space V, and suppose that p is the smallest positive integer for which TP = To. Prove the following results. (a) N(T) CN(TI+) for every positive integer i. a (b) There is a sequence of ordered bases B1, B2, ..., 3 such that B; is a basis for N(T) and Bi+1 contains 3. for 1 <i<p-1. (c) Let B = B, be the ordered basis for N(Tº) = V in (b). Then (T) is an upper triangular matrix with each diagonal entry equal to zero. (d) The characteristic polynomial of T is (-1)"4". Hence the charac- teristic polynomial of T splits, and 0 is the only eigenvalue of T.