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5. Let T R3 > R3 be the linear transform defined by the following properties: T(0,0,1) = (0,0,0), If v is in the ry-plane, then v is reflected across the x + y = 0 plane There is a matrix A such that T(x) = Ax. The goal of this problem is to understand A. (a) (3 points) Find a basis {u, v, w} where the action of T is well-understood. Give also T(u), T(v) and T(w) (b) (3 points) Find the eigenvalues of A and a basis for each eigenspace of A. (Think geometrically.) (c) (3 points) What is A? You may express it as product of matrices and their inverses. single matrix. (Think geometrically. (d) (3 points) What is A2? Give it explicitly as a

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