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

The operator polynomial collapses to the identity

For a variable x, consider the $R -$ vector space V = { $a_{0} + a_{1} x + a_{2} x^{2}$ | $a_{1}, a_{2}, a_{3} \in R$ }. 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?

A. $(T^{3} - 3 T^{2} + 3 T)^{2} 025 (x) = x$ ✓
B. $(T^{3} - 3 T^{2} + 3 T)^{2} 025 (x) = x + 1$
C. $(T^{3} - 3 T^{2} + 3 T)^{2} 025 (x) = 2025! x$
D. $(T^{3} - 3 T^{2} + 3 T)^{2} 025 (x) = 2025! x + 1$

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.

See pricing

50 are analysed free — try those first.

Related counterexample: Equivalent matrices are similar

The operator polynomial collapses to the identity

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