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Part CCSIR NET December 2024the-kernel-is-generated-by-the-minimal-polynomial

The kernel is generated by the minimal polynomial

For each n > 1, let V be the R−\mathbb{R}-vector space of all n × n real matrices and A ∈ V be invertible. Consider the R−\mathbb{R}-linear transformation φ:R[x]→V\varphi : \mathbb{R}[x] \to V such that φ(xk)=Ak\varphi(x^k) = A^k for all k ≥ 1 and φ(1)\varphi(1) is the identity matrix of order n. Which of the following statements are necessarily true?

  1. A.φ\varphi is one-to-one but not onto.
  2. B.φ\varphi is onto but not one-to-one.
  3. C.There exists f∈R[x]f \in \mathbb{R}[x] such that deg f ≤ n and ker⁡φ=\ker \varphi = {fg | g∈R[x]g \in \mathbb{R}[x]}.✓ Correct
  4. D.deg h ≥ n for every nonzero h∈ker⁡φh \in \ker \varphi.

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 execution slip.

See pricing

50 are analysed free — try those first.

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

Related counterexample: Same characteristic polynomial ⇒ similar

More on this topic

The chapter behind this: Characteristic vs minimal polynomial — what each tells you — free to read

From Eigenvalues and Canonical Forms › Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

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