MATH 220 Name MIDTERM EXAMINATION ID October 19 2004 Section There are 20 multiple choice questions Each problem is worth 5 points 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 20 PROBLEMS ON 11 PAGES INCLUDING THIS ONE MATH 220 MIDTERM EXAMINATION PAGE 2 1 Which of the following matrices is in reduced echelon form a b c d 1 A 0 0 0 B 1 0 1 C 0 0 1 D 0 0 5 1 1 2 0 1 1 0 0 0 0 0 1 2 0 0 0 1 0 4 1 5 0 0 2 0 4 6 2 Let A Find the general solution to the homogeneous equation Ax 0 1 1 3 2 a b c d x1 2x3 3x4 x2 x3 5x4 x x are free 3 4 x1 2x3 3x4 x2 6x3 5x4 x x are free 3 4 x1 x3 x4 x2 x3 x4 x x are free 3 4 x1 7x4 x2 x3 x4 x x are free 3 4 MATH 220 MIDTERM EXAMINATION 3 Consider the linear system solution set a b c d x1 x2 x3 2 4x1 3x2 5 PAGE 3 Find the parametric vector form of its 1 3 x 3 x3 4 0 1 1 3 x x3 3 4 2 3 5 4 x x3 0 1 1 3 x 3 x3 4 0 1 1 2 1 4 Consider the following vectors v1 1 v2 1 v3 3 For what value s of h is 2 1 h the following set v1 v2 v3 linearly dependent a h 4 b h 4 c All real numbers h 6 0 d h 0 MATH 220 MIDTERM EXAMINATION PAGE 4 1 0 2 1 5 Which of the following vectors is NOT in Span 0 0 a b c d 1 2 0 10 5 0 1 2 1 0 0 0 6 Let T R2 R2 be the linear transformation that first rotates vectors of R2 counterclockwise through 90 about the origin then reflects vectors about the line y x Find the standard matrix of T a b c d 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 1 MATH 220 MIDTERM EXAMINATION PAGE 5 1 1 1 2 2 7 Let u v and let T R R be a linear transformation such that T u 1 0 2 2 and T v Find the image of 3u v under the transformation T 5 a b c d 1 1 2 3 1 7 2 1 1 1 1 8 If T is a linear transformation whose standard matrix is given by A 0 2 0 then 1 2 1 which 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 MATH 220 MIDTERM EXAMINATION 9 Which of the following set of vectors is linearly independent a b c d 1 5 4 2 0 9 2 1 3 6 5 10 2 3 1 2 1 1 1 1 0 1 0 1 1 1 0 1 1 0 1 0 10 Let A 1 1 a 1 b 2 c 3 d 4 0 1 1 0 How many rows of A contain a pivot position 0 1 2 1 PAGE 6 MATH 220 MIDTERM EXAMINATION PAGE 7 1 1 3 11 Let A 1 3 5 The solution set of the homogeneous equation Ax 0 is 0 1 1 0 a Only the trivial solution x 0 0 b A line through the origin c A plane through the origin d All R3 12 Let T R5 R2 be a linear transformation Which of the following statements is always true for such transformations a T is onto b T is one to one c The standard matrix of T is a 2 5 matrix d The standard matrix of T is a 5 2 matrix MATH 220 MIDTERM EXAMINATION 13 Which of the following transformations is linear a b c d x1 4 T x2 x1 x2 3 x1 x1 x2 T x2 x1 x2 x1 3x1 T x2 x1 5x2 x1 sin x1 T x2 x1 x2 5 7 14 The inverse of the matrix A is 2 3 a b c d 3 7 2 5 3 7 2 5 3 7 2 5 3 7 2 5 PAGE 8 MATH 220 MIDTERM EXAMINATION 1 0 2 0 1 15 If A and B 4 1 What is BA 3 1 5 2 0 a b c d 4 17 2 5 4 2 5 2 5 0 0 1 0 1 1 1 0 2 1 1 0 3 1 1 0 2 2 1 1 16 If A 4 6 0 then Ax 0 has 2 7 2 a only one solution b two solutions c infinitely many solution d No solution PAGE 9 MATH 220 MIDTERM EXAMINATION PAGE 10 17 Suppose A is a 4 5 matrix such that each row contains a pivot Which of the following statements is false a The columns of A span R4 b Ax 0 has a free variable c The columns of A are linearly independent d The columns of A are linearly dependent 1 3 18 Let T R R be a linear transformation If T e1 and T 2e1 e2 what is 2 5 the standard matrix of T 2 a b c d 1 2 1 2 1 2 3 2 2 3 5 1 1 1 0 1 4 MATH 220 MIDTERM EXAMINATION PAGE 11 1 0 4 b1 19 Let A 2 1 6 and b b2 Then Ax b is consistent if 1 3 2 b3 a b1 b2 b3 0 b 2b1 b2 b3 0 c 5b1 3b2 b3 0 d b1 b2 b3 0 20 Which of the following statements is false a Every homogeneous linear system is consistent b If A is a 3 2 then the linear transformation defined by T x Ax cannot be onto c If A is a 2 3 then the linear transformation defined by T x Ax cannot be onto d Two nonzero vectors in Rn are linearly dependent if one of the vectors is a scalar multiple of the other