Homework 9, due 10/19/2015
1. Read Artin §2.10 (“The Correspondence Theorem”) Note that the main
theorem of this section is the version for groups of what you proved for
vector spaces in HW7.
2. Artin 2.10.3
3. Artin 2.10.5
4. In this exercise, assume the theorem on existence of Jordan canonical form,
even though we haven’t proven it. Let T : V → V be a linear operator on
a vector space V over a field F . Assume F contains all eigenvalues of T , so
that we can apply the theorem on Jordan canonical form. Now moreover
assume that dimF (V ) = 5, and that T has only one eigenvalue, λ, so that
V is equal to the λ-generalized eigenspace Vλgen . In each of the following
parts, I will list possibilities for dim ker((T − λ)i ) for various i: your task
is to determine the Jordan canonical form of the operator T under these
dimension assumptions.
(a) dim ker(T − λ) = 5.
(b) dim ker(T − λ) = 4, dim ker((T − λ)2 ) = 5.
(c) dim ker(T − λ) = 2, dim ker((T − λ)2 ) = 4, dim ker((T − λ)3 ) = 5.
(d) dim ker((T − λ)i ) = i for i = 1, 2, . . . , 5.
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