Question: Exercise I (30 + 10 = 40 points) Let A = 8 1 2] 1. Find the singular value decomposition, A = UEVT. 2. Deduce

Exercise I (30 + 10 = 40 points) Let A = 8 1 2] 1. Find the singular value decomposition, A = UEVT. 2. Deduce an orthonormal basis for each of the four fundamental subspaces of A. Exercise II (5 x 10 = 50 points) Are the following transformations linear? If so, provide a complete proof. If not, explain why. 1. (x1, 12, 23) = (21 - 12, 12 + 23). 2. T : R3 - R', where T(21, 12, 23) = (21 + 13, -212). 3. T : R3 - R&, where T(r1, 12, 13) = (201 - 13, 12 + 23, 0, 21 - 312). 4. T : P2 -> Pa, where T[p(x)] = xp(x) + (x - 1)2p"(x). 5. T(x1, 12, 23) = (21 - 12, 12 + 23, 1 + 13) Exercise III (10 points) Find the change of basis matrix that changes S'-coordinates into -coordinates (both are bases for 2)

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