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

Invariant complements need semisimplicity

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

  1. A.There exists a subspace W of Cn\mathbb{C}^{n} such that Cn=V+W\mathbb{C}^{n} = V + W and V ∩ W = {0}.✓ Correct
  2. B.There exists a subspace W of Cn\mathbb{C}^{n} such that T(W)⊆W,Cn=V+WT(W) \subseteq W, \mathbb{C}^{n} = V + W and V ∩ W = {0}.
  3. C.Suppose that there exists a positive integer k such that Tᵏ is the identity map. Then there exists a subspace W of Cn\mathbb{C}^{n} such that T(W)⊆W,Cn=V+WT(W) \subseteq W, \mathbb{C}^{n} = V + W and V ∩ W = {0}.✓ Correct
  4. D.Suppose that there exists a subspace W of Cn\mathbb{C}^{n} such that T(W)⊆W,Cn=V+WT(W) \subseteq W, \mathbb{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, and why this one tests hypothesis dropped.

See pricing

50 are analysed free — try those first.

The trap it tests

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Drill statements like this

Related counterexample: Diagonalisable ⇒ invertible

More on this topic

The chapter behind this: Diagonalisability — the criteria card — free to read

From Eigenvalues and Canonical Forms › Diagonalisability criteria

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