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Part CCSIR NET December 2025every-power-of-an-idempotent-collapses-to-itself-so-w-is-really-just-span-of-i-and-t

Every power of an idempotent collapses to itself so w is really just span of i and t

Let V be a real vector space and L(V) denote the space of linear operators on V. Let T∈L(V) be a non-zero operator such that T2=TT^{2}=T. Consider the subspace W of L(V) spanned by {I,Tn:nI, T^{n} : n is a positive integer}. Which of the following statements are necessarily true?

  1. A.The set {S∈W:S2=SS\in{}W : S^{2}=S} contains exactly 2 elements.
  2. B.dimR(W)=2dim_\mathbb{R}(W) = 2.
  3. C.If U∈L(V) is such that U2=UU^{2}=U and (T+U)2=T+U(T+U)^{2}=T+U, then TU=0.✓
  4. D.If U∈L(V) is such that U2=UU^{2}=U and (T−U)2=T−U(T-U)^{2}=T-U, then (TU)2=)^{2}=TU.✓

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.

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50 are analysed free — try those first.

Related counterexample: Equivalent matrices are similar

More on this topic

The chapter behind this: Linear transformations, matrices and similarity — free to read

From Vector Spaces and Linear Maps › Linear transformations, matrix representation, change of basis

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