Let T: C7 → C7 be a C-linear operator with eigenvalues 2, 3 and 5. Consider the subspace W := {v ∈ C7 : (T - 5I)kv = 0 for some integer k > 0} of C7. Suppose that (T - 2I)2(T - 3I)2(T - 5I)2 = 0. Which of the following statements is necessarily true?

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Mathematics · Linear Algebra

Verified Solution4 OptionsUpdated: October 2026👁️1 View

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Correct answer

ker((T - 2I)2025) = ker((T - 2I)2026)

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Answer 1

(T - 2I)(T - 3I) is a nilpotent operator

Answer 2

ker((T - 2I)2025) = ker((T - 2I)2026)

Answer 3

dim W ≥ 2

Answer 4

T has at least four linearly independent eigenvectors

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