A linear transformation T maps A(2, 1) to A'(4, 1). If the transformation matrix T has the form [1 k; 0 1], find T and the images of B(3, 2), C(3, 4), D(2, 3) under T.

a) Step 1: Determine the transformation matrix T. The transformation T is a shear with the x-axis invariant. A shear matrix with the x-axis invariant has the form 1 & k \\ 0 & 1 . Given A(2, 1) is mapped to A'(4, 1). 1 & k \\ 0 & 1 2 \\ 1 = 4 \\ 1 This gives two equations: 1(2) + k(1) = 4 2 + k = 4 k = 2 0(2) + 1(1) = 1 1 = 1 (consistent) So, the transformation matrix T is: T = 1 & 2 \\ 0 & 1 Step 2: Determine the coordinates of B', C', and D'. The original coordinates are B(3, 2), C(3, 4), D(2, 3). For B': 1 & 2 \\ 0 & 1 3 \\ 2 = 1(3) + 2(2) \\ 0(3) + 1(2) = 3 + 4 \\ 2 = 7 \\ 2 So, B' is (7, 2). For C': 1 & 2 \\ 0 & 1 3 \\ 4 = 1(3) + 2(4) \\ 0(3) + 1(4) = 3 + 8 \\ 4 = 11 \\ 4 So, C' is (11, 4). For D': 1 & 2 \\ 0 & 1 2 \\ 3 = 1(2) + 2(3) \\ 0(2) + 1(3) = 2 + 6 \\ 3 = 8 \\ 3 So, D' is (8, 3). b) Step 1: Determine the transformation matrix H. Let the transformation matrix H be a & b \\ c & d . Given A'(4, 1) is mapped to A''(0, 1) and B'(7, 2) is mapped to B''(-12, -15). For A': a & b \\ c & d 4 \\ 1 = 0 \\ 1 This gives: 1) 4a + b = 0 2) 4c + d = 1 For B': a & b \\ c & d 7 \\ 2 = -12 \\ -15 This gives: 3) 7a + 2b = -12 4) 7c + 2d = -15 From (1), b = -4a. Substitute into (3): 7a + 2(-4a) = -12 7a - 8a = -12 -a = -12 a = 12 Substitute a = 12 into b = -4a: b = -4(12) = -48 From (2), d = 1 - 4c. Substitute into (4): 7c + 2(1 - 4c) = -15 7c + 2 - 8c = -15 -c + 2 = -15 -c = -17 c = 17 Substitute c = 17 into d = 1 - 4c: d = 1 - 4(17) = 1 - 68 = -67 So, the transformation matrix H is: H = 12 & -48 \\ 17 & -67 Step 2: Determine the coordinates of C''. C' is (11, 4) from part (a). 12 & -48 \\ 17 & -67 11 \\ 4 = 12(11) + (-48)(4) \\ 17(11) + (-67)(4) = 132 - 192 \\ 187 - 268 = -60 \\ -81 So, C'' is (-60, -81). Send me the next one 📸