Homework 19

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1. Exercise 7.A.14
2. Exercise 7.A.17.
   Hints for this one (Hopefully I haven't made any typos):
   For the first part, start by showing that null T ⊂ null T ^k, which is fairly straightforward. Showing that null T ^k ⊂ null T takes a little more effort. One way is to show that T ^*T ^k-1v=0 for any v ∊ null T ^k (do so by showing its length is 0, for instance). Then use that and normality to show that ⟨T ^k-1v,T ^k-1v⟩=0 (what does that imply?). Then argue that this eventually leads to v ∊ null T.
   For the second part, show that range T ^k ⊂ range T and then show they have the same dimension (why is that enough?).