MATH 220 Name FINAL EXAMINATION DECEMBER 16 2004 ID Section There are 25 multiple choice questions Four possible answers are given for each problem only one of which is correct When you solve a problem note the letter next to 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 THE USE OF CALCULATORS DURING THE EXAMINATION IS FORBIDDEN PLEASE SHOW YOUR PSU ID CARD TO YOUR INSTRUCTOR WHEN YOU FINISH GOOD LUCK CHECK THE EXAMINATION BOOKLET BEFORE YOU START THERE SHOULD BE 25 PROBLEMS ON 14 PAGES INCLUDING THIS ONE MATH 220 FINAL EXAMINATION PAGE 2 1 2 0 5 1 2 0 5 0 1 Consider the matrices A 0 0 1 3 and B 0 0 1 3 0 Which of the following 0 0 0 1 0 0 0 0 1 statements is true a None of the matrices is in reduced echelon form b Both A and B are in reduced echelon form c Only A is in reduced echelon form d Only B is in reduced echelon form 2x1 x2 x3 10 x2 x3 4 2 What is the general solution of the linear system x1 2x2 3x3 5 x1 x2 a x3 is free x1 x3 b x2 is free x1 x2 c x3 is free 3 2x3 4 x3 1 2x2 4 x2 3 x3 4 x3 d None of the above MATH 220 FINAL EXAMINATION PAGE 3 3 2 3 If T is a linear transformation whose standard matrix is given by A 1 0 then which 5 1 of the following statements is true a T is one to one but not onto b T is not one to one but it is onto c T is both one to one and onto d T is neither one to one nor onto 1 2 5 b1 4 Let A 0 2 6 and b b2 Then Ax b is consistent if 1 3 10 b3 a 2b1 5b2 2b3 0 b 2b1 b2 5b3 0 c b1 3b2 b3 0 d b1 b2 b3 0 MATH 220 FINAL EXAMINATION 5 Find the parametric vector form of the solution set of the system PAGE 4 x1 x2 2x3 5 2x1 3x2 4x3 2 1 a x x3 1 2 13 10 b x 8 x3 8 0 1 13 1 1 c x x2 8 x3 0 2 d None of the above 1 1 1 6 Let v1 1 v2 4 v3 1 Which of the following statements is true 1 1 2 a Span v1 v2 v3 is the origin b Span v1 v2 v3 is a line through the origin c Span v1 v2 v3 is a plane through the origin d Span v1 v2 v3 is all of R3 MATH 220 FINAL EXAMINATION PAGE 5 7 Consider the two planes given by 2x1 3x2 5x3 7 and x1 x2 5x3 1 Which of the following statements is true a Their intersection is empty b Their intersection is the point 2 1 0 2 c Their intersection is a line through the point 2 3 1 with direction u 1 0 d None of the above statements describes the intersection of these planes 8 Let A be the standard matrix of a linear transformation T Rn Rn that is onto Which of the following statements is true a A is not invertible b Ax 0 has a only the trivial solution c The columns of A are linearly dependent d T is not one to one MATH 220 FINAL EXAMINATION PAGE 6 9 Suppose T is the linear transformation that first rotates points through radians counter4 clockwise and then projects points onto the x axis The standard matrix of T is 2 2 a 0 2 b 2 0 0 c 2 2 2 2 0 2 2 0 0 2 2 d None of the above 10 Which of the following sets is a subspace of R3 a b c d x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 7 x2 x3 0 sin x2 x3 0 x21 x22 0 MATH 220 FINAL EXAMINATION 1 2 3 4 1 4 2 What is the dimension of the null space of A 11 Let A 1 1 2 5 3 a 1 b 2 c 3 d 4 2 3 0 2 12 Let A 1 0 3 1 Find a basis for the column space of A 0 2 4 0 2 a 1 0 3 2 1 0 b 0 2 3 0 2 c 1 0 3 0 2 4 d None of the above PAGE 7 MATH 220 FINAL EXAMINATION 1 1 1 1 2 Which of the following vectors is in the null space of A 13 Let A 0 1 5 11 a b c d 0 1 1 1 1 0 1 2 1 1 2 1 1 2 3 14 What is the characteristic polynomial of A 0 1 2 0 2 1 a 1 1 3 b 1 2 1 c 1 2 d None of the above PAGE 8 MATH 220 FINAL EXAMINATION 0 3 15 The determinant of A 0 0 2 0 0 0 0 1 0 1 PAGE 9 0 0 is 4 4 a 24 b 24 c 0 d None the above 16 Let A and B be 3 3 matrices such that det A 4 and det B 3 Find det 2A 1 B 2 a 9 2 b 72 c 18 d 18 MATH 220 FINAL EXAMINATION PAGE 10 1 5 3 17 Find the eigenvalues of A 0 3 0 0 2 2 a 1 2 3 b 0 1 2 c 1 3 5 d 2 3 5 2 1 18 Let T R R be a linear transformation whose standard matrix is A Find 1 2 a basis B such that the B matrix of T is diagonal 2 a B b B c B d B 2 1 1 0 1 0 1 1 1 1 2 1 1 1 1 1 1 MATH 220 FINAL EXAMINATION 19 Which of the following sets is an orthonormal set a b c d 1 1 1 0 1 1 2 3 1 3 2 3 1 3 2 6 1 3 1 3 1 6 1 3 1 6 0 1 1 0 1 0 2 1 1 0 1 1 20 If A Then A3 equals 1 1 0 3 1 2 29 a 28 24 b 25 29 c 28 56 55 56 55 56 57 d None of the above PAGE 11 MATH 220 FINAL EXAMINATION PAGE 12 0 1 1 2 1 1 1 21 Let B Find the coordinate vector of x 2 relative to the 1 1 1 3 3 orthogonal basis B for R 1 6 a 5 2 2 3 1 6 b 5 2 2 1 6 c 5 2 2 3 d None …