2. Consider a fixed matrix M R2x2, and define the Lyapunov operator LM (Rx2, R) (Rx2, R) by LM(Q)=MTQ+QM, QERX2 (a) Show that LM is a linear operator. -61 find A, the representation of LM with respect to the basis [9], (b) For the special case M = 1 = ez = =89 3 = = [], 2=6 1], =[%], 1 e4 = (c) Compute the change of basis matrix mapping the representation of a vector with respect to the basis {e..4} to its representation with respect to the basis {4}, defined as 69 4 = -681 (d) Use MATLAB to compute A = PAP-1, the representation of the Lyapunov operator with respect to the basis {e..4}. 1Revised January 23, 2023. (e) With M as given in (2b), find bases for the range and nullspace of LM. (Recall that these subspaces lie in Rx2.)