Part BCSIR NET December 2024the-span-of-commutators-is-all-of-sl

The span of commutators is all of sl

Let V be the $R -$ vector space of 5 × 5 real matrices. Let S = {AB − BA | A, B ∈ V} and W denote the subspace of V spanned by S. Let $T : V \to R$ be the linear transformation mapping a matrix A to its trace. Which of the following statements is true?

A.W = ker(T)✓
B.W ⊊ ker(T)
C.W ∩ ker(T) ⊊ W
D.W ∩ ker(T) ⊊ ker(T)

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: AB = I ⇒ BA = I for all matrices

The span of commutators is all of sl

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