The commutator of two linear transformations L, M: V †’ V on a vector space V is defined as
K = [L, M) = L —‹ M - M —‹ L. (7.15)
(a) Prove that the commutator K is a linear transformation.
(b) Prove that L and M commute if and only if [L, M] = 0.
(c) Compute the commutators of the linear transformations defined by the following pairs of matrices:

(d) Prove the Jacobi identity
[[L, M], N] + [[N, L], M]
+ [[M, N], L] = O (7.16)
is valid for any three linear transformations.
(e) Verify the Jacobi identity for the first three matrices in part (c).
(f) Prove that the commutator B(L, M) = [L, M] defines a bilinear map on L(V, V), cf. Exercise 7.1.18.

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