Solved: Linear transformations Consider the linear

Chapter 13, Problem 60

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Linear transformations Consider the linear transformation T in _2 given by x = au + bv, y = cu + dv, where a, b, c, and d are real numbers, with ad _ bc. a. Find the Jacobian of T. b. Let S be the square in the uv-plane with vertices 10, 02, 11, 02, 10, 12, and 11, 12, and let R = T1S2. Show that area1R2 = _ J1u, v2 _. c. Let / be the line segment joining the points P and Q in the uv-plane. Show that T1/2 (the image of / under T) is the line segment joining T1P2 and T1Q2 in the xy-plane. (Hint: Use vectors.) d. Show that if S is a parallelogram in the uv-plane and R = T1S2, then area1R2 = _ J1u, v2 _ area1S2. (Hint: Without loss of generality, assume the vertices of S are 10, 02, 1A, 02, 1B, C2, and 1A + B, C2, where A, B, and C are positive, and use vectors.)