Exercises 8. T:R² → R² defined by T(x1,x2) = (ı+x2, x1 – x2 – 1). 9. T:R? → R° defined by T(x1,x2) = (1, x1+ x2, 2.x2) 1. Compute ax + by for each of the following: (а) а %— 2, b %3D —-3, г %3 (2, 4) and y 3 (3, —1). (b) а %3D 10, b %3 -2, х %3 (1,0, —1) and y 3D (2, -4, 3). (c) a = 5, b = -1, x = (4, 2, –1, 1) and y = (-1,3,5, 7). 10. Determine which of the following maps are linear maps. If the map is linear give the matrix associated to the linear map. Explain your reasoning. (») Lı : R* -→ R° where La;)-(****) x + y +3 2y +1 2. Let x = (4, 7) and y = (2, –1). Write the vector ax + By sin x R' where L2 x + y 2y as a vector in coordinates. (b) L2 :R² (c) L3 : R? → R where L3 = x + y 3. Let x = (1, 2), y = (1, –3), and z = (-2, – 1). Show that you can write z = ax + By for some a, BER. 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. (;). (:) Aej = 3 Ae2 = and Aez = 4. Can the vector z = (2, 3, –1) be written as 12. The cross product of two 3-vectors x = y = (y1, Y2, Y3) is the 3-vector (x1, x2, x3) and z = ax + By where x = (2, 3, 0) and y = (1, –1, 1)? x x y = (x2y3 – x3y2, –(x1y3 – x3y1), x1y2 – x2y1). Let K = (2,1, –1). 5. Let x = (3, –2), y = (2, 3), and z = (1,4). For which real numbers a, B, y does (a) Show that the mapping L : R → R defined by ах + Ву + yz%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 x 3 matrix A such that L(x) = Ax, 6. T: R* → R² defined by T(x1,x2, x3) = (x1+2x2-x3, X1- 4.x3). that is, L = LA. 7. T: R² → R² defined by T(x1, x2) = (x1 +x1x2, 2x2). 61