Math 2360 Exam III 18 November 2009Form BSection I For each of the problems in the this section at least one of the choices is correct. For some of theproblems more than one of the choices is correct. Record your answers for problems in thissection on the answer sheet (last page). 1. (9 pts) Determine whether the following are linear transformations from to 43a. b. 1342412422()224xxxxxLxxxx12423411`1 2 34424()xxxxxxLxxxxxxxxc. 0() 10Lx2. (9 pts) Determine whether the following are linear transformations from to 4P4Pa. b. 2(()) () () ()Lpx px xp x xp x () ( )(())pxpxLpxxc. (()) (0) (0) ()Lpx p p pxSection II Continue the answers for these problems on the answer sheet. Append additional pages as needed. You do not need to rewrite the problem statements on your answer sheets. Work carefully. Do yourown work. Show all relevant supporting steps!3. (10 pts) Consider the linear transformation mapping to given by 4212341234() [ 3 2 4 , 2 2 ]TLxx xxxxxx    xa. Find the kernel of L.b. Find the dimension of the range of L.4. (10 pts) Consider the linear transformation mapping to given by4P4P01(()) ()xLpx ptdtxa. Find the kernel of L.b. Find the dimension of the range of L.5. (8 pts) Consider the linear transformation mapping to given by 431234123424( ) [ 2 3 4 ,2 4 3 ,2 2 ]TLxxxxxxxxxx     xFind the standard matrix representation for L.6. (16 pts) Consider the linear transformation mapping to given by 32123123() [ 2 3 ,2 3 2 ]TLxxxxxx   xFind a matrix A which represents L with respect the standard basis in and the123[, , ]eee3ordered basis in where .12[, ]bb21210and21bb7. (10 pts) Find the equation of the plane in which is normal to the vector and which3[1, 2, 3]TNpasses through the point .0[3,2,2]TP8. (10 pts) Find the distance in from the point to the plane given by30[1, 2,3]T P.22 6xyz9. (10 pts) Let S be the subspace in which is spanned by the set . Find41, 2, 1, 1 , 1, 2, 1, 1TT  a basis for .S10. (10 pts) Find the least squares solution of the linear system x11 111 221 210 1xName _________________________ Form BAnswers1. i. Yes No ii. Yes No iii. Yes No2.i. Yes No ii. Yes No iii. Yes