Math 2360 Section I 1 6 pts Final Exam Form A 8 8 8 For each of the problems in the this section at least one of the choices is correct For some of the problems more than one of the choices is correct Record your answers for problems in this section on the answer sheet last page 1 2 0 3 Consider the following matrix A 2 1 3 3 Which of the following matrices are 1 2 1 2 row equivalent to A a d 2 8 pts 1 2 0 1 0 0 2 0 3 1 3 3 0 1 2 0 0 3 1 0 1 0 1 2 1 b 0 0 1 0 e 1 2 0 3 1 3 3 0 1 2 2 0 3 3 3 9 2 1 2 1 c 0 0 1 0 f 0 0 0 7 1 0 2 0 1 5 0 0 2 1 1 3 0 0 0 Consider the following four matrices 3 0 1 A 1 2 1 2 7 1 B 3 4 0 1 0 D 2 3 1 1 0 1 C 2 1 Which of the following arithmetic operations are possible 3 8 pts a 3A 2B b AB e C 2 f DA 2 2 c AC D d B g D 2 A2 h DB BD 3 Consider the following subsets of a x1 x2 x3 T x1 0 or x2 0 b x1 x2 x3 T x1 0 or x2 0 or x3 0 c x1 x2 x3 T x1 x3 0 d x1 x2 x3 T x12 x2 2 x3 2 1 Which of the above subsets of are subspaces of 3 3 4 8 pts Consider the following subsets of P4 a p P4 p 1 0 b p P4 p 1 p 0 c p P4 p is even d p P4 deg p odd Which of the above subsets of P4 are subspaces of P4 5 8 pts Consider the following subsets of 3 a 2 1 1 2 0 3 c 1 5 1 0 3 6 0 2 4 b 0 1 1 2 2 5 0 2 3 d 1 3 1 0 1 1 2 1 0 1 1 0 Which of the above subsets of are spanning sets for 3 6 8 pts 3 Consider the following subsets of 3 a 1 1 1 1 0 2 c 0 2 1 1 1 2 0 2 3 b 1 2 6 0 1 2 1 2 4 d 1 2 2 3 0 1 2 1 1 0 1 1 Which of the above subsets of are linearly independent 3 7 4 pts Consider the set S 1 x 1 x 1 2 x x 2 which is a subset of P3 Is the set S linear independent 8 8 pts 3 Consider the following subsets of a 1 1 1 1 0 1 c 1 1 3 0 1 2 1 3 6 b 0 2 5 2 2 1 3 2 4 d 1 1 1 2 0 1 3 0 1 0 1 1 Which of the above subsets of forms a basis for 3 9 9 pts 3 Determine whether the following are linear transformations from 4 to 3 x1 4 x2 4 x4 x1 x2 a L x 2 x1 2 x2 x3 b x4 1 x2 L x 0 x4 1 x x1 c L x x1 x2 x1 x2 x3 10 9 pts Determine whether the following are linear transformations from P3 to P4 x a L p x x 2 p x c 11 6 pts b L p x p t dt 0 L p x x 1 p x Consider the sets of vectors given below Which sets are orthogonal sets 1 2 2 a 2 2 1 2 3 0 c 1 b 1 2 2 1 3 3 0 2 1 5 2 1 0 3 1 12 3 4 4 2 4 T T 12 8pts Which of the following sets forms an orthonormal basis for 2 1 1 T T a 1 2 2 1 5 5 c 13 9 pts T 1 2 1 2 1 2 1 2 T b 3 5 4 5 3 5 4 5 d 1 0 0 1 T T T T Let A be a 4 4 matrix Which the following are always true a dim N A dim R AT b dim N A dim R A c dim R A dim R AT d N A R AT e N A R A f R A R AT g 4 N A R AT h 4 N A R A i 4 R A R AT Section II Answer the problems in this section on separate paper You do not need to rewrite the problem statements on your answer sheets Work carefully Do your own work Show all relevant supporting steps 14 9 pts i ii iii a 15 8 pts Each of the following augmented matrices is in row echelon form For each matrix indicate whether the corresponding system of linear equations is consistent or inconsistent For each case in which the corresponding system of linear equations is consistent indicate whether the system has a unique solution or infinitely many solutions For each case in which the corresponding system of linear equations is consistent and has a unique solution find that unique solution 1 1 0 1 0 0 0 0 1 0 0 0 2 1 1 1 0 1 0 0 1 1 0 1 b 0 0 0 0 1 0 0 1 1 1 1 0 0 1 0 0 1 1 0 1 c 0 0 0 0 1 0 0 1 1 1 1 2 0 0 0 0 The following augmented matrix is in reduced row echelon form Find the solution set of the corresponding system of linear equations 1 0 a 0 0 0 2 0 1 2 1 1 0 2 1 0 0 1 2 1 0 0 0 0 0 16 9 pts 17 9 pts For each of the following pairs of matrices find an elementary matrix E such that EA B 1 0 2 a A 1 1 1 2 1 2 1 0 2 B 2 1 2 1 1 1 1 2 0 2 2 2 1 3 b A 1 2 1 4 0 1 2 3 1 2 0 2 2 0 3 3 B 1 2 1 4 0 1 2 3 1 2 0 2 2 2 1 1 1 3 A Consider the matrix A given by 1 2 1 0 2 2 2 3 4 9 1 2 A straightforward 1 0 reduction by elimination shows that A is row equivalent to U where 1 0 U 0 0 0 1 0 0 0 4 3 0 7 3 0 1 3 0 2 3 1 2 0 2 0 0 1 0 a Find a basis for the row space of A b Find a basis for …