Math 2360 Exam III 30 July 2008Form A - Make UpSection 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 this section onthe answer sheet (last page). 1. (3 pts) Let A be a matrix. If the dimension of the null space of A is 4, then the rank of A is:610×a. 2 b. 4 c. 6 d. 8e. Undeterminable from the given information2. (9 pts) Determine whether the following are linear transformations from to 3¡2¡a. b. 12324()1xxxLx++=+x13213221()22xxxLxxx−+=+−xc. 21()xLx=x3. (9 pts) Determine whether the following are linear transformations from to 3P3Pa. b. ()(0)(())pxpLpxx−=(())()(1)()Lpxpxxpx′=−−c. ()()(())2pxpxLpx+−=Section IIAnswer the problems in this section on separate paper. You do not need to rewrite the problemstatements on your answer sheets. Work carefully. Do your own work. Show all relevant supportingsteps!4. (15 pts) Consider the matrix A given by . A straightforward reduction by121212313101311A−−−=−−−−−elimination shows that A is row equivalent to U where .105010131100000U−=−−a. Find a basis for the row space of A.b. Find a basis for the column space of A.c. Find a basis for the null space of A.5. (6 pts) Consider the linear transformation mapping to given by 3¡4¡3212321321()(3,2,24,23)TLxxxxxxxxxx=−−−−++xFind the standard matrix representation for L.6. (10 pts) Consider the linear transformation mapping to given by 3¡3¡12312123()(22,,2)TLxxxxxxxx=+−−−+xa. Find the kernel of L.b. Find the dimension of the range of L.7. (10 pts) Consider the linear transformation mapping to given by3P3P2(())2()()Lpxpxxpx′′=−a. Find the kernel of L.b. Find the dimension of the range of L.8. (10 pts) Consider the linear transformation mapping to which satisfies the conditions that3P3P22(1)32Lxxxx+−=++2(1)32Lxxx−+=+2()3Lxx+=Find the value of 2(34)Lxx+−9. (12 pts) Consider the linear transformation mapping to given by 3¡2¡12323()(21,2)TLxxxxx=−++xFind a matrix A which represents L with respect the standard basis in and the ordered123[,,]eee3¡basis in where .12[,]bb2¡1212and21bb==−−10. (16 pts) Consider the linear operator on given by 2¡1212()(3,2)TLxxxx=+−xFind a matrix A which represents L with respect to the standard basis .12[,]eeFind a matrix B which represents L with respect the ordered basis where 12[,]bb.1222and12bb==Name _________________________ Form BAnswers1. ______2. i. Yes No ii. Yes No iii. Yes No3.i. Yes No ii. Yes No iii. Yes