This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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To verify if a transformation T: R^2 R^2 defined by T(x,y) = (2x, 2y) is linear, we must check two properties: 1. Additivity: T(u + v) = T(u) + T(v) for any vectors u, v R^2. 2. Homogeneity: T(cu) = cT(u) for any vector u R^2 and any scalar c R. Step 1: Check the additivity property. Let u = (x_1, y_1) and v = (x_2, y_2) be two arbitrary vectors in R^2. First, calculate T(u + v): u + v = (x_1 + x_2, y_1 + y_2) T(u + v) = T(x_1 + x_2, y_1 + y_2) = (2(x_1 + x_2), 2(y_1 + y_2)) = (2x_1 + 2x_2, 2y_1 + 2y_2) Next, calculate T(u) + T(v): T(u) = T(x_1, y_1) = (2x_1, 2y_1) T(v) = T(x_2, y_2) = (2x_2, 2y_2) T(u) + T(v) = (2x_1, 2y_1) + (2x_2, 2y_2) = (2x_1 + 2x_2, 2y_1 + 2y_2) Since T(u + v) = T(u) + T(v), the additivity property holds. Step 2: Check the homogeneity property. Let u = (x, y) be an arbitrary vector in R^2 and c be an arbitrary scalar. First, calculate T(cu): cu = (cx, cy) T(cu) = T(cx, cy) = (2(cx), 2(cy)) = (2cx, 2cy) Next, calculate cT(u): T(u) = T(x, y) = (2x, 2y) cT(u) = c(2x, 2y) = (c · 2x, c · 2y) = (2cx, 2cy) Since T(cu) = cT(u), the homogeneity property holds. Since both the additivity and homogeneity properties are satisfied, the transformation T is linear. The transformation T(x,y) = (2x, 2y) is indeed linear. That's 2 down. 3 left today — send the next one.