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Part CCSIR NET December 2024the-commutator-is-the-derivative-itself

The commutator is the derivative itself

For a variable x, consider the real vector space V = {ax3+^{3} + bx2+^{2} + cx + d | a,b,c,d∈Ra, b, c, d \in \mathbb{R}}. Let D : V → V be the linear transformation where D(f) is the derivative of f with respect to x, and M : V → V be the linear transformation M(f) = xD(f). Which of the following statements are true?

  1. A.DM ≠ MD.✓ Correct
  2. B.D + M is invertible.
  3. C.DM is invertible.
  4. D.rank(DM) = rank(MD).

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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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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