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Extracted text: Exercises 8. T: R? → R² defined by T (x1,x2) = (x1+x2, x1 – X2 – 1). 9. T: R² → R³ defined by T(x1,x2) = (1, x1+x2,2x2) 1. Compute ax + by for each of the following: 10. Determine which of the following maps are linear maps. = (3, –1). = (2, –4, 3). (c) a = 5, b = –1, x = (4, 2, – 1, 1) and y = (-1, 3, 5, 7). (а) а — 2, b — -3, х — (2, 4) and y = If the map is linear give the matrix associated to the linear (b) а — 10, b 3 — 2, х — (1,0, —1) and y — map. Explain your reasoning. ;)-(+1 x + y + 3 2y + 1 (a) L1 : R? → R² where L1 ( 2. Let x = (4, 7) and y = (2, –1). Write the vector ax + By sin x as a vector in coordinates. (b) L2 : R² –→ R° where L2 x + Y 2y (;) 3. Let x = (1, 2), y = (1, –-3), and z = (-2, – 1). Show that (c) L3 : R² –→ R where L3 = x + y you can write z = ax + By for some a, ß E R. 11. Find the 2 x 3 matrix A that satisfies Hint: Set up a system of two linear equations in the un- knowns a and B, and then solve this linear system. (:). 2 Aei = Ae2 and Аез 3 -1 4. Can the vector z = (2, 3, – 1) be written as 12. The cross product of two 3-vectors x = (x1, 12, x3) and z = ax + By y = (y1, Y2, Y3) is the 3-vector where x = (2, 3,0) and y = (1, –1, 1)? тху3 (х2уз — г:3у2, — (х1уз — «зу1), х192 — 22у1). Let K = (2,1, -1). : (3, —2), у — (2, 3), and z %3D B,y does 5. Let x = (1,4). For which real (a) Show that the mapping L : R³ → R³ defined by numbers a, ат + By + 72 3 (1,—2)? L(x) = x x K is a linear mapping. In Exercises 6 – 9 determine whether the given transformation is linear. (b) Find the 3 × 3 matrix A such that L(x) = Ax, 6. T: R* → R² defined by T(x1, x2, x3) = (xı+2x2-x3,X1¬- 4x3). that is, L = LẠ. 7. T: R² → R² defined by T(x1,x2) = (x1 +x1£2,2x2). 61