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Part BCSIR NET December 2024the-operator-polynomial-collapses-to-the-identity

The operator polynomial collapses to the identity

For a variable x, consider the R−\mathbb{R}-vector space V = {a0+a1x+a2x2a_{0} + a_{1}x + a_{2}x^{2} | a1,a2,a3∈Ra_{1}, a_{2}, a_{3} \in \mathbb{R}}. Let T : V → V be the linear transformation defined by T(f) = f + df/dx, where df/dx denotes the derivative of f with respect to x. Which of the following statements is true?

  1. A.(T3−3T2+3T)2025(x)=x(T^{3} - 3T^{2} + 3T)^2025(x) = x✓ Correct
  2. B.(T3−3T2+3T)2025(x)=x+1(T^{3} - 3T^{2} + 3T)^2025(x) = x + 1
  3. C.(T3−3T2+3T)2025(x)=2025!x(T^{3} - 3T^{2} + 3T)^2025(x) = 2025! x
  4. D.(T3−3T2+3T)2025(x)=2025!x+1(T^{3} - 3T^{2} + 3T)^2025(x) = 2025! x + 1

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 standard counterexample.

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50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

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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