QUESTION 4 Kernel of linear transformation Choose one . 5 points Let T : R? -+ 12 such that T(r, y) = (x - y, 2x + y). Find ker(T) and dim(ker(T)) O ker (T) = o, and dim(ker (T)) = 0 O ker(T) = 3, 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: R2 -> IR? such that T I ) = (I + 3y I + 4y. Find Im(T) and dim(Im(T)) O Im(T) = span . (), (2, and dim(Im(T)) = 2 O Im(T) = span {(3. (,and dim( Im(T) = 2 O Im(T) = span (1 , and dim( Im(T) = 1 QUESTION 6 Translation in 3D Space Choose one . 5 points Write the translation matrix in the direction of 3 = (1, 2, 3) O 0 1 \\ To(1,2,3) = 0 1 3 0 2 O 0 0 0 2 0 0 0 3 0 0 0 O 0 0 0 2 To(1,2,3) = 1 3 0 1QUESTION 1 Select the mappings that are linear transformations Choose all that apply . 5 points Select the mappings that are linear transformations O T : R + R such that T(z) = 21 T : R' + R such that T(z, y) = zy T : R' + R such that T(2) = At, where A = ((6 7) O T : R -+ R such that T(z) =13 GROUP 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' + R3 such that T ( ) = (32 + 6y O 171 = (2 3) O 17 = (3 8) O QUESTION 2.2 Computing linear transformation standard matrix Choose one . 5 points Find the standard matrix of the following linear transformation : /2x+ y+z T: R' + R' such that T = I+2y -z cty+ 3z O 0 [7 = 0 1 0 0 1 O 2 1 IT = 3 O [7] =GROUP 3 Linear Transformation Matrix from Non-Standard Basis Group . 2 questions Let T : RR> > R' such that T(x, 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 = {v1, 12} = {(2, 1), (3, 2)} and Bu = {un, w2} = {(1, 1), (1, 2)} That is : [T] B-,B. = [Mw | Mo] O ]B., B. = 3 6 O ] B., B. = 1 11/3 -2 O QUESTION 3.2 Computing a vector image using transition matrix Choose one . 5 points Using previous question 3.1 answer for [Ilg_,B.. calculate [ula, if [ulBy = ()using [UB =[TB,B. B. O Be = (3 ) O -28/3 Be = -1 OQUESTION 7 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 u = (1, 2, 3) from the previous question 6? O O 10 Or p = 10 O D = Or p QUESTION 8 Inverse Translation matrix Choose one . 5 points What is the inverse matrix of the translation in the direction of U = (1, 2,3) from the previous question 6? O T-: 1 O 0 -3 O 0 0 1/2 O O 0 -1 0 T-= 0 1 -3 O 0 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) = */2(radian) O OL R,(1/2) = 0 O -1 R,(1/2) = 0 O 0 1 0 RE(1/2) = -1 0 0 0 1 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 0 R, 1(1/2) = 0 0 0 0 O 0 -1 0 R, ' (1/2) = 0 0 O 0 1 R, 1 (#/2) = -1 0 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 P O QUESTION 12 Scaling Matrix in 3D Space . Choose one . 5 points Write the scaling matrix with scaling factors 8 = 4. 8, = 2 and 8. = 1 O 1 1 2 1 O 0 2 0 4 O 0 20 1 QUESTION 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 8, =4 , a, = 2 and 8, = 1 O 13 O OQUESTION 14 Inverse Scaling Matrix Choose one . 5 points What is the inverse matrix of the scaling matrix with scaling factors 87=4 , 8 = 2 and 8, = 1 O 1 0 1/4 0 1 1/2 0 1 O 1 S= 1/2 0 90 1/4 O 1/4 0 S= 0 1/2 0 0 QUESTION 15 Linear Operators Choose all that apply . 5 points Which of the following linear transformations are linear operators ? OF : R3 + 13 such that T(z, y) = (1 - 2y, y) T: R3 + R such that T(z, y) = z - 2y OT: R3 + 13 such that T(z, y) = (x - y,y, z + y) OF: R3 + R' such that T(z, y, 2) = (x -2y - zy,Ity + 2) QUESTION 16 Composition of Linear operators Choose one . 5 points Let Ti : R' + 123 such that T ( ) = () PUB 72 : R' + R such that T ( ) = ( 2z + 3y) Ity ) Find Tio Tz O [Tion] - (2 1) and TioTh : R3 + R such that Tio Th 2r + y O Tion : R? + R3 such that Tio TY ( ) = (3x + 5y) O Tion : R' + R such that TioTy ( I ) = (4x + 5y) I + 2yQUESTION 17 w One-to-One Linear Operator Choose all that apply . 5 points Select the linear operators that are one-to-one 0 T : R' + R such that T ( ) - (y) 0 7 : R' + R such that T (#) = (2+9) 0 T : R + R such that T 0 7 : R + R auch that T = 6x + 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-I O det ( T]) =0 = T is not invertible O det (|7]) =3 = Tis invertible T: R' + R where T- () =1/3\\+2y I -y O det ( 7) ) = 1 = Tis invertible T-1: R + where T-1 (I =1 -1 + 2y