MATH 220 NAME FINAL EXAMINATION STUDENT NUMBER MAY 2 2006 INSTRUCTOR FORM A SECTION NUMBER This examination will be machine processed by the University Testing Service Use only a number 2 pencil on your answer sheet On your answer sheet identify your name this course Math 220 and the date Code and blacken the corresponding circles on your answer sheet for your student I D number and class section number Code in your test form FIVE POINTS WILL BE DEDUCTED FROM YOUR FINAL SCORE IF YOU DO NOT FILL IN YOUR ID NUMBER SECTION NUMBER OR TEST VERSION CORRECTLY There are 25 multiple choice questions each worth six points For each problem four possible answers are given only one of which is correct You should solve the problem note the letter of the answer that you wish to give and blacken the corresponding space on the answer sheet Mark only one choice darken the circle completely you should not be able to see the letter after you have darkened the circle Check frequently to be sure the problem number on the test sheet is the same as the problem number of the answer sheet THE USE OF A CALCULATOR CELL PHONE OR ANY OTHER ELECTRONIC DEVICE IS NOT PERMITTED DURING THIS EXAMINATION CHECK THE EXAMINATION BOOKLET BEFORE YOU START THERE SHOULD BE 25 PROBLEMS ON 14 PAGES INCLUDING THIS ONE MATH 220 FINAL EXAMINATION FORM A PAGE 2 1 Which of the following is the coefficient matrix for a homogeneous system Ax 0 with only the trivial zero solution a b c d 1 A 0 0 1 A 0 0 1 A 0 0 1 A 0 0 0 0 2 1 0 0 0 0 3 0 0 0 1 0 0 0 1 0 2 3 0 1 0 0 2 3 4 5 0 6 2 Find the condition that b1 b2 b3 must satisfy in order for the linear system x1 x2 2x3 b1 x1 2x2 3x3 b2 to be consistent 3x1 4x2 7x3 b3 a b3 2b1 5b2 b b3 3b2 c b3 2b1 b2 d The system is consistent for any b1 b2 b3 MATH 220 FINAL EXAMINATION FORM A PAGE 3 3 Suppose u v w is a linearly independent set of vectors in R3 Which of the following sets is also linearly independent a u u v u w b 0 v c u u v u w v w d u u v v 1 0 3 4 Let A 2 1 0 then the span of the columns of A is 0 2 2 a 0 b a line c a plane d all of R3 MATH 220 FINAL EXAMINATION FORM A PAGE 4 5 Suppose S R2 R2 is given by S x1 x2 0 0 and T R3 R3 is given by T x1 x2 x3 x1 2x2 x3 3 then which of the following is true a Neither S nor T is a linear map b S is a linear map but T is not c S is not a linear map but T is d Both S and T and linear maps 2 2 6 Let T R R be a linear map Suppose that T 3 What is T 4 a b c d 1 5 8 7 2 1 17 8 2 1 3 1 2 and T 4 2 1 MATH 220 FINAL EXAMINATION FORM A PAGE 5 7 Find the standard matrix of the linear map T R2 R2 which first expands in the x1 direction by a factor of two and in the x2 direction by a factor of three then reflects across the x2 axis a b c d 3 1 0 2 3 0 0 2 2 1 0 3 2 0 0 3 1 3 1 0 2 3 8 Suppose A B C Find A2 BC 2AT 0 1 2 2 1 0 a b c d 5 3 0 9 1 1 0 9 5 6 6 9 11 12 7 3 MATH 220 FINAL EXAMINATION FORM A PAGE 6 4 2 0 3 2 is an invertible matrix What is the second column of A 1 9 Given that A 5 1 1 0 a b c d 0 1 0 1 3 1 1 1 4 0 0 0 0 1 2 10 Let A be an n n matrix and suppose the equation Ax b is inconsistent then which of the following statements is true a b is in the column space of A b A is row equivlant to In c A has less than n pivot columns d AT is invertible MATH 220 FINAL EXAMINATION FORM A 11 Which of the following is a subspace of R3 x1 a x2 x1 x2 x 3 b The x1 x2 plane 2 c The point 1 0 1 0 d The set of vectors of the form 1 t 0 for any scalar t 0 0 12 Let A be an m n matrix Which of the following is always true a Rank A m b The column space of A is a subspace of Rn c The null space of A has dimension n d Rank A dim Nul A n PAGE 7 MATH 220 FINAL EXAMINATION FORM A 5 2 0 13 Let A 1 3 1 then what is det A 4 2 0 a 18 b 18 c 12 d 12 4 0 5 7 6 3 14 Let A 1 2 1 and B 1 2 1 Which of the following is true 7 6 3 4 0 5 a det A 0 b det B 2 c det A det B d det A det B PAGE 8 MATH 220 FINAL EXAMINATION FORM A PAGE 9 1 0 1 15 Find the least squares solution x of Ax b where A 1 3 and b 4 1 6 5 a b c d 4 x 2 2 x 1 1 x 1 4 3 x 2 3 16 Let A be an n n matrix Which of the following is NOT always true a The characteristic equation of A is of the form p 0 where p is a degree n polynomial b If A is similar to B then A and B have the same eigenvalues c A has at most n distinct eigenvalues d If 1 is an eigenvalue of A then A is invertible MATH 220 FINAL EXAMINATION FORM A 3 2 17 Let A 1 1 a b c d 1 0 0 2 PAGE 10 1 1 2 3 2 1 2 noting that What is A3 1 3 1 1 1 3 5 17 4 22 32 0 15 2 13 42 7 22 16 32 8 16 18 Let B be a basis for R3 and let T R3 R3 be a linear map such that b1 b2 b3 2 1 3 T B 1 1 0 What is T b1 b2 b3 0 1 2 a 3b3 b b1 b2 b3 c 3b1 b2 b3 d 2b1 MATH 220 FINAL EXAMINATION FORM A PAGE 11 19 Let A …