MATH 220 MIDTERM EXAMINATION October 21, 2003Name ID # 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 BEFOREYOU START. THERE SHOULD BE ?? PROBLEMSON ?? PAGES (INCLUDING THIS ONE).MATH 220 MIDTERM EXAMINATION PAGE 21. What relation must b1, b2and b3satisfy to ensure that the following system of equations isconsistent?2x1− 4x2+ 10x3= b14x1− 5x2+ 8x3= b2−2x1+ x2+ 2x3= b3a) −b1+ b2+ b3= 0b) 7b1+ b2− b3= 0c) b1+ b2+ b3= 0d) All values of b1, b2and b32. Which of the following matrices is not in echelon form?a)1 0 0 00 0 1 20 0 0 0b)0 1 2 00 0 1 30 0 0 1c)3 4 5 60 6 7 00 0 8 1d)2 4 6 80 0 9 100 2 3 4MATH 220 MIDTERM EXAMINATION PAGE 33. If A =4 −6−8 126 −9, then which o f the following is a nontrivial solution of t he equationAx = 0?a)12b)32c)23d)214. If T : R3→ R2is a linear transformation whose standard matrix is1 4 −53 −7 4, thenwhich of the following statements is true?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 t o one, but it is onto.MATH 220 MIDTERM EXAMINATION PAGE 45. If A =1 3 −51 4 −8−3 −7 9, then which of the following best describes t he geometric form of theset of all solutions of Ax = 0?a) It is the zero vector.b) A line.c) A plane.d) 3-dimensional space.6. If1 1 3 21 2 4 31 3 5 his the augmented matrix for a system of linear equations, then for whichvalue(s) of h is the system consistent?a) h = 1b) h = 2c) h = 4d) h 6= 2MATH 220 MIDTERM EXAMINATION PAGE 57. 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 ofsolutions.b) If A and B are n × n matrices such that the columns of A span Rnand B is rowequiva lent 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 R5ontoR6.d) If A is an m × n matrix, then the linear transformation x 7→ Ax is a one-to-onemapping.8. If A =1 4 3−1 −2 02 2 3, then what is the second row of A−1?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 69. If T : R2→ R3is a linear transformation such that T (e1+ e2) =123and T (e1−e2) =301,then what is the standard matrix for T ?Here e1=10and e2=01.a)1 32 03 1b)2 −11 12 1c)3 10 21 3d)1 0 10 1 210. If T : R27→ R3is a linear transformation such that T (u) =12and T (v) =−32, thenwhat is T (2u −3v)?a)11−2b)42c)−24d) There is not enough information to compute T (2u − 3v) .MATH 220 MIDTERM EXAMINATION PAGE 711. Which of the following sets of vectors is linearly independent?a)11−1,234,000b)2−13,1−21,305c)678,410,200d)153,666,51312. If T : R27→ R2is the linear transformation that is obtained by first rotating points by π/2 inthe counterclockwise direction and then reflecting points about the line x2= x1, then whatis the standard matrix for T ?a)−1/√2 1/√21/√2 1/√2b)1/√2 1/√2−1/√2 1/√2c)1/√2 1/√21/√2 −1/√2d)1/√2 −1/√21/√2 1/√2MATH 220 MIDTERM EXAMINATION PAGE 813. For what va lue o f h is the vector2h1in the span of t he vectors12−1and−366?a) h = 1b) h = 4c) h = 8d) h = 1614. If A =2 3 21 −2 −1and B =1 00 1−1 0, then what is (AB)T?a)0 32 −1b)2 3 21 −2 −1−2 −3 −2c)0 23 −2d)2 1 −23 −2 −32 −1 −2MATH 220 MIDTERM EXAMINATION PAGE 915. If A, B and C are n ×n matrices, then which o f the fo llowing statements is not always true?a) A(B + C) = AB + ACb) If AB = BA, then (A + B)2= A2+ 2AB + B2.c) (AB)T= BTAT.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−1b) 7Bc)17Bd)17B−1MATH 220 MIDTERM EXAMINATION PAGE 1017. Which of the following formulas defines a linear transformation from R2to R2?a) Tx1x2=5x23x1b) Tx1x2=x1+ 1x2+ 2c) Tx1x2=cos x1sin x2d) Tx1x2=x21x218. If T : R27→ R3is defined by the formulaTx1x2=2x1− 8x2−4x1+ hx2x1+ (12 − h)x2,then for what value(s) of h is T one-to-one?a) All real h.b) h = 16c) No value of hd) h 6= 16MATH 220 MIDTERM EXAMINATION PAGE 1119. If A is an m × n matrix that has a pivot position in every row, then which of the followingstatements 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.20. If A =k 2−3 4, then for what value of k is A not invertible?a) 2b) 3/2c) −2d)