QUESTION 1 V Select the mappings that are linear transformations Choose all that apply . 5 points Select the mappings that are linear transformations T : R - R such that T(x) = 21 T : R' -+ R such that T(I, y) = ry T : R' - R' such that T(I) = Ar, where A = T : R - R such that T(x) = 13GROUP 2 Linear Transformation Standard Matrix from Standard Basis Group . 2 questions QUESTION 2.1 Computing linear transformation standard matrix Choose one . 5 points Find the standard matrix of the following linear transformation : T : R' -+ R' such that T (I = 2x + 3y 5x + 6y O ] = CT N 6 O [T] = OQUESTION 2.2 Computing linear transformation standard matrix Choose one . 5 points Find the standard matrix of the following linear transformation : I 2ctytz T : R - R such that T I + 2y - 2 cty + 32 O 0 0 [T] = 0 1 0 0 0 1 O [T] = 2 -1 O [T] = 3GROUP 3 Linear Transformation Matrix from Non-Standard Basis Group . 2 questions Let T : R' -> R' such that T(r, y) = (x + y, x - 2y) QUESTION 3.1 Computing linear transformation matrix from non-standard basis Choose one . 5 points Find the matrix of T relative to the basis By = (01, U2) = {(2, 1), (3, 2) } and Bu = {w1, w2} = {(1, 1), (1,2) } That is : [TIB_,R. = [Mc | Mo] O [TIB.,B. = -3 11 -6 O [TB., B. = 11/3 -2 OQUESTION 3.2 w Computing a vector image using transition matrix Choose one . 5 points Using previous question 3.1 answer for [T], B , calculate [u]g, if [ula = ( using [1 B. = [TB.,B. . [u]B. O O -28/3 -1 O [u]B =Kernel of linear transformation Choose one . 5 points Let T: IR' + R such that T(r, y) = (x - y, 2x + y). Find ker() and dim(ker (T)) O ker (T) = 6, and dim(ker(T)) = 0 O ker ( T) = 6, and dim(ker (T)) = 1 O ker (T) = span {(1, 0)}, and dim(ker (T)) = 1 QUESTION 5 Range or image of a linear transformation Choose one . 5 points Let T: R? > > such that T ( X ) _ ( 2+ 3y Find Im(T) and dim(Im(T)) O Im ( T) = span ((1) . ()and dim ( Im(7) =2 O Im(T) = span {(3) . (4and dim( Im(7) = 2 O Im(T) = span and dim( Im (T)) = 1QUESTION 6 Translation in 3D Space Choose one . 5 points Write the translation matrix in the direction of v = (1, 2,3) O 01 0 3 0 2 O 0 0 0 0 0 0 0 0 0 O 1 0 1 \\ 0 1 0 2 0 01 3 0 0 0 1QUESTION 7 V Image of a point by translation Choose one . 5 points What is the image of p = (4,5, 2) after the translation in the direction of 1 = (1, 2, 3) from the previous question 6? O O 4 10 = 10 6 6 O = Or pQUESTION 8 V Inverse Translation matrix Choose one . 5 points What is the inverse matrix of the translation in the direction of v = (1, 2,3) from the previous question 6? O L T- -2 0 -3 O 0 1/2 T = 0 1/3 0 O HO OHOO 0QUESTION 9 Rotation Matrix in 3D Space Choose one . 5 points Write the rotation matrix about the z-axis with angle 90 where cos(90) =0 and sin(90) = 1. note that: 90(degree) = x/2(radian) O R= (1/ 2 ) = -1 0 0 O R= (1/2) = 10 0 0 O R= (7/2) = -10 0 QUESTION 10 Inverse of a Rotation Matrix Choose one . 5 points Write the inverse rotation matrix about the z-axis with angle 90 where cos(90) =0 and sin(90) = 1 O -1 R. ' (1/2) = 0 0 0 0 O -1 0 0 0 0 1 O 1 0 R. ' (W/2) = -10 0 0 1QUESTION 11 image of a point by rotation Choose one . 5 points What is the image of p = (1, -1, 0) after a rotation of 90 about the z-axis? O O O QUESTION 12 Scaling Matrix in 3D Space . Choose one . 5 points Write the scaling matrix with scaling factors $, = 4, $y = 2 and $, = 1 O 1 0 4 S42,1 = 0 1 0 1 O 0 0 2 0 0 4 O 4 0 0 S42,I = 2 0 0QUESTION 13 Image of a Point by a Scaling Matrix Choose one . 5 points What is the image of p = (1, 2,3), after it has been scaled by S, = 4 , Sy = 2 and $: = 1 13 O QUESTION 14 Inverse Scaling Matrix Choose one . 5 points What is the inverse matrix of the scaling matrix with scaling factors Sx = 4 , Sy = 2 and $z = 1 O 0 1/4 S- 1 1/2 0 1 O 1/2 0 0 1/4 O 1/4 0 0 S= 1/2 0 0QUESTION 15 Linear Operators Choose all that apply . 5 points Which of the following linear transformations are linear operators ? T: R' + 1R2 such that T(z, y) = (1 - 2y, y) OT: R2 + IR such that T(z, y) = 1 - 2y T: R3 + 1RX3 such that T(z, y) = (1 - y,y, z + y) T: 13 + R3 such that T(z, y, 2 ) = (1 - 2y - z, y, z ty + 2) QUESTION 16 Composition of Linear operators Choose one . 5 points Let Ti : IR3 + IR such that T ( Z ) = (2+ 23) and T2 : 12 + 183 such that T ( X ) = ( 2x + 33) Find Ti o T2 [Ti Til = (3 1) and TioTe : R? _+ 123 such that TioTz (y ) = (27+3) O (2 :) TioT? : IR? > IR? such that TioTi ( I ) = (3x + 5y O (1 2 ) andQUESTION 17 One-to-One Linear Operator Choose all that apply . 5 points Select the linear operators that are one-to-one O Ti : R' + 13 such that T (7 ) = (x) 0 Ti : R' _+ 12 such that T ( ) = (193) Ti : R3 + 182 such that T ( ) - (23 Ti : R3 + 1R2 such that T ( ) = 2x + y 61 + 3y QUESTION 18 Inverse of a one-to-one Linear Operator Choose one . 5 points Verify if the linear operator T is invertible, and compute its inverse T-1 T : IR3 + 13 where T ( I ) = (2 +3) O det ([T]) = 0 = T is not invertible O det ([T]) = 3 = T is invertible T-1 : 13 + 1R23 where T-1 (7 ) =1/3 (-2+ 29) O det ([T]) = 1 = T is invertible T-1 : R' _ 123 where T' (7 ) = 13