Part CCSIR NET December 2024invariant-complements-need-semisimplicity

Invariant complements need semisimplicity

For every integer n ≥ 2, consider $a C -$ linear transformation $T : C^{n} \to C^{n}$ . Let V be a subspace of $C^{n}$ such that T(V) ⊆ V. Which of the following statements are necessarily true?

A.There exists a subspace W of $C^{n}$ such that $C^{n} = V + W$ and V ∩ W = {0}.✓
B.There exists a subspace W of $C^{n}$ such that $T (W) \subseteq W, C^{n} = V + W$ and V ∩ W = {0}.
C.Suppose that there exists a positive integer k such that Tᵏ is the identity map. Then there exists a subspace W of $C^{n}$ such that $T (W) \subseteq W, C^{n} = V + W$ and V ∩ W = {0}.✓
D.Suppose that there exists a subspace W of $C^{n}$ such that $T (W) \subseteq W, C^{n} = V + W$ and V ∩ W = {0}. Then there exists a positive integer k such that Tᵏ is the identity map.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

See pricing

50 are analysed free — try those first.

Related counterexample: Diagonalisable ⇒ invertible

Invariant complements need semisimplicity

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