Mathematics · Linear Algebra
For every integer n ≥ 2, consider a &mathbb;C-linear transformation T: &mathbb;Cn → &mathbb;Cn. Let V be a subspace of &mathbb;Cn such that T(V) ⊆ V. Which of the following statements is necessarily true?
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- 1If there exists a subspace W of &mathbb;Cn such that T(W) ⊆ W, &mathbb;Cn = V + W and V ∩ W = {0}, then there exists a positive integer k such that Tk is the identity map.
- 2If there exists a positive integer k such that Tk is the identity map, then there exists a subspace W of &mathbb;Cn such that T(W) ⊆ W, &mathbb;Cn = V + W and V ∩ W = {0}.✓
- 3There exists a subspace W of &mathbb;Cn such that T(W) ⊆ W, &mathbb;Cn = V + W and V ∩ W = {0}.
- 4There exists a subspace W of &mathbb;Cn such that &mathbb;Cn = V + W and V ∩ W = {0}.
Correct answer
If there exists a positive integer k such that Tk is the identity map, then there exists a subspace W of &mathbb;Cn such that T(W) ⊆ W, &mathbb;Cn = V + W and V ∩ W = {0}.
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