The commutator of two linear transformations $L, M: V \rightarrow V$ on a vector space $V$ is
$$
K=[L, M]=L \circ M-M \circ L \text {. }
$$
(a) Prove that the commutator $K$ is a linear transformation on $V$. (b) Explain why Exercise 1.2 .30 is a special case. (c) Prove that $L$ and $M$ commute if and only if $[L, M]=\mathrm{O} . \quad(d)$ Compute the commutators of the linear transformations defined by the following pairs of matrices:
(i) $\left(\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right),\left(\begin{array}{rr}-1 & 0 \\ 1 & 2\end{array}\right)$,
(ii) $\left(\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right),\left(\begin{array}{rr}1 & 0 \\ 0 & -1\end{array}\right)$,
(iii) $\left(\begin{array}{lll}1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1\end{array}\right), \quad\left(\begin{array}{rrr}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{array}\right)$.
(e) Prove that the Jacobi identity
$$
[[L, M], N]+[[N, L], M]+[[M, N], L]=\mathrm{O}
$$
is valid for any three linear transformations. $(f)$ Verify the Jacobi identity for the first three matrices in part $(c)$. $(g)$ Prove that the commutator $B[L, M]=[L, M]$ defines a bilinear $\operatorname{map} B: \mathcal{L}(V, V) \times \mathcal{L}(V, V) \rightarrow \mathcal{L}(V, V)$ on the Cartesian product space, cf. Exercise 7.1.18.