Let L be the line that is the graph of y = x in the xy - plane , R2. Let RL be t

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Let L be the line that is the graph of y = x in the xy - plane , R2. Let RL be the reflection through L, let PL be the projection onto L, and let Theta be a counterclockwise rotation through 90 degree about the origin. The mappings RL, PL, and Theta are linear operators on R2. Let epsilon = {e1, e2} be the standard basis for R2, i.e. = [1 0] and = [0 1]. Find the matrix representation of RL with respect to epsilon, find the eigenvalues of RL and each eigenspace, and compute the determinate of RL. Find the matrix representation of PL with respect to epsilon, find the eigenvalues of PL and each eigenspace, and compute the determinate of PL. Find the matrix representation of Theta with respect to epsilon, find the eigenvalues of Theta and a basic for each eigenspace, and compute the determinate of Theta.

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