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

A. $φ$ is one-to-one but not onto.
B. $φ$ is onto but not one-to-one.
C.There exists $f \in R [x]$ such that deg f ≤ n and $ker φ =$ {fg | $g \in R [x]$ }.✓
D.deg h ≥ n for every nonzero $h \in ker φ$ .

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: Same characteristic polynomial ⇒ similar

The kernel is generated by the minimal polynomial

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