MATH 220 Name MIDTERM EXAMINATION ID October 21 2003 Section There are multiple choice questions in this examination Each problem has four choices Blacken only ONE oval for each problem Each problem is worth 5 points THE USE OF CALCULATORS DURING THE EXAMINATION IS FORBIDDEN CHECK THE EXAMINATION BOOKLET BEFORE YOU START THERE SHOULD BE PROBLEMS ON PAGES INCLUDING THIS ONE MATH 220 MIDTERM EXAMINATION 1 What relation must b1 b2 and b3 satisfy consistent 2x1 4x1 2x1 to ensure that the following system of equations is 4x2 10x3 b1 5x2 8x3 b2 x2 2x3 b3 a b1 b2 b3 0 b 7b1 b2 b3 0 c b1 b2 b3 0 d All values of b1 b2 and b3 2 Which of the following matrices is not in echelon form a b c d 1 0 0 0 0 0 3 0 0 2 0 0 0 0 0 0 1 2 0 0 0 1 2 0 0 1 3 0 0 1 4 5 6 6 7 0 0 8 1 4 6 8 0 9 10 2 3 4 PAGE 2 MATH 220 MIDTERM EXAMINATION 4 3 If A 8 6 Ax 0 a b c d PAGE 3 6 12 then which of the following is a nontrivial solution of the equation 9 1 2 3 2 2 3 2 1 1 4 5 4 If T R R is a linear transformation whose standard matrix is then 3 7 4 which of the following statements is true 3 2 a T is one to one and onto b T is one to one but not onto c T is neither one to one nor onto d T is not one to one but it is onto MATH 220 MIDTERM EXAMINATION PAGE 4 1 3 5 4 8 then which of the following best describes the geometric form of the 5 If A 1 3 7 9 set of all solutions of Ax 0 a It is the zero vector b A line c A plane d 3 dimensional space 1 1 3 6 If 1 2 4 1 3 5 value s of h a h 1 b h 2 c h 4 d h 6 2 2 3 is the augmented matrix for a system of linear equations then for which h is the system consistent MATH 220 MIDTERM EXAMINATION PAGE 5 7 Which of the following statements is not always true a If a linear system of equations has two solutions then it has an infinite number of solutions b If A and B are n n matrices such that the columns of A span Rn and B is row equivalent to A then the columns of B also span Rn c If A is a 6 5 matrix then the linear transformation x 7 Ax does not map R5 onto R6 d If A is an m n matrix then the linear transformation x 7 Ax is a one to one mapping 1 4 3 8 If A 1 2 0 then what is the second row of A 1 2 2 3 a 1 2 0 b 1 2 1 2 1 2 c 1 4 1 4 1 4 d 0 1 0 MATH 220 MIDTERM EXAMINATION PAGE 6 1 3 2 3 9 If T R R is a linear transformation such that T e1 e2 2 and T e1 e2 0 3 1 1 0 then what is the standard matrix for T Here e1 and e2 0 1 a b c d 1 2 3 2 1 2 3 0 1 1 0 3 0 1 1 1 1 1 2 3 0 1 1 2 3 1 then and T v 10 If T R 7 R is a linear transformation such that T u 2 2 what is T 2u 3v 2 3 11 a 2 4 b 2 2 c 4 d There is not enough information to compute T 2u 3v MATH 220 MIDTERM EXAMINATION PAGE 7 11 Which of the following sets of vectors is linearly independent 2 0 1 1 3 0 a 1 4 0 1 3 2 b 1 2 0 3 1 5 4 2 6 7 1 0 c 8 0 0 6 5 1 d 5 6 1 3 6 3 12 If T R2 7 R2 is the linear transformation that is obtained by first rotating points by 2 in the counterclockwise direction and then reflecting points about the line x2 x1 then what is the standard matrix for T a b c d 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 MATH 220 MIDTERM EXAMINATION 2 1 3 2 and 6 13 For what value of h is the vector h in the span of the vectors 1 1 6 a h 1 b h 4 c h 8 d h 16 1 0 2 3 2 14 If A and B 0 1 then what is AB T 1 2 1 1 0 0 3 a 2 1 2 3 2 b 1 2 1 2 3 2 0 2 c 3 2 2 1 2 d 3 2 3 2 1 2 PAGE 8 MATH 220 MIDTERM EXAMINATION PAGE 9 15 If A B and C are n n matrices then which of the following statements is not always true a A B C AB AC b If AB BA then A B 2 A2 2AB B 2 c AB T B T AT d If AB 0 then A 0 or B 0 16 If A is a 3 3 invertible matrix and the inverse of 7A B then what is the inverse of A a 7B 1 b 7B c 1 B 7 d 1 1 B 7 MATH 220 MIDTERM EXAMINATION 17 Which of the following formulas defines a linear transformation from R2 to R2 a b c d x1 5x2 T x2 3x1 x1 x1 1 T x2 x2 2 x1 cos x1 T x2 sin x2 2 x1 x T 1 x2 x2 18 If T R2 7 R3 is defined by the formula 2x1 8x2 x1 T 4x1 hx2 x2 x1 12 h x2 then for what value s of h is T one to one a All real h b h 16 c No value of h d h 6 16 PAGE 10 MATH 220 MIDTERM EXAMINATION PAGE 11 19 If A is an m n matrix that has a pivot position in every row then which of the following statements is always true a The columns of A span Rn b The columns of A span Rm c The columns of A are linearly independent d The columns of A are linearly dependent k 2 20 If A then for what value of k is A not invertible 3 4 a 2 b 3 2 c 2 d 3 2