MATH 220 MIDTERM EXAMINATION October 19, 2004Name ID # Section #There are 20 multiple choice questions. Each problem is worth 5 points. Four possibleanswers 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 spaceon the answer sheet. Mark only one choice; darken the circle completely (you shouldnot 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 BEFOREYOU START. THERE SHOULD BE 20 PROBLEMSON 11 PAGES (INCLUDING THIS ONE).MATH 220 MIDTERM EXAMINATION PAGE 21. Which of the following matrices is in reduced echelon f orm?a) A =1 5 10 1 20 0 1b) B =0 1 01 0 00 0 0c) C =1 1 20 0 00 0 1d) D =1 0 40 1 50 0 02. Let A =2 0 4 6−1 1 −3 2. Find the general solution to the homogeneous equation Ax = 0.a)x1= −2x3− 3x4x2= x3− 5x4x3, x4are freeb)x1= 2x3− 3x4x2= 6x3− 5x4x3, x4are freec)x1= x3− x4x2= x3+ x4x3, x4are freed)x1= 7x4x2= x3− x4x3, x4are freeMATH 220 MIDTERM EXAMINATION PAGE 33. Consider the linear system:x1+ x2− x3= 24x1+ 3x2= 5.Find the pa rametric vector form of itssolution set.a) x =−130+ x3−341b) x =−13+ x33−4c) x =250+ x33−41d) x =−1−30+ x33414. Consider the following vectors v1=1−12, v2=211v3=−13h. Fo r what value(s) of h isthe following set {v1, v2, v3} linearly dependent?a) h = 4b) h = −4c) All real numbers h 6= 0d) h = 0MATH 220 MIDTERM EXAMINATION PAGE 45. Which of the following vectors is NOT in Span(120,010)?a)120b)1050c)121d)0006. Let T : R2→ R2be the linear transformation that first rotates vectors of R2counterclockwisethrough 90◦about the origin, then reflects vectors about the line y = −x. Find the standardmatrix of T .a)1 −11 0b)−1 00 1c)0 −11 0d)1 10 1MATH 220 MIDTERM EXAMINATION PAGE 57. Let u =11, v =10, and let T : R2→ R2be a linear transformation such that T (u) =12and T (v) =25. Find the image of 3u − v under the transformation T .a)11b)23c)17d)218. If T is a linear transformation whose standard matrix is given by A =1 1 10 2 0−1 2 1, thenwhich 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 ontoc) T is both one-to-one and ontod) T is neither one-to-one nor ontoMATH 220 MIDTERM EXAMINATION PAGE 69. Which of the following set of vectors is linearly independent?a)12,50,49b)135,2610c)120,311,2−11d)110,011,101.10. Let A =1 0 10 1 01 0 −11 2 1. How many rows of A contain a pivot position?a) 1b) 2c) 3d) 4MATH 220 MIDTERM EXAMINATION PAGE 711. Let A =1 1 31 3 50 1 1. The solution set of the homogeneous equation Ax = 0 isa) Only the trivial solution x =000b) A line through the origin.c) A plane through the origin.d) All R3.12. Let T : R5→ R2be a linear transformation. Which of the following statements is alwaystrue for such tr ansformations.a) T is ontob) T is one-to-onec) The standard matrix of T is a 2 × 5 matrixd) The standard matrix of T is a 5 × 2 matrixMATH 220 MIDTERM EXAMINATION PAGE 813. Which of the following transformations is linear?a) Tx1x2=4x1− x2b) Tx1x2=x31− x2x1− x2c) Tx1x2=3x1x1− 5x2d) Tx1x2=sin(x1)x1− x214. The inverse of the matrix A =5 72 3isa)3 72 5b)−3 7−2 5c)3 −7−2 5d)3 72 5MATH 220 MIDTERM EXAMINATION PAGE 915. If A =2 0 13 1 5and B ==1 04 −12 0. What is BA?a)4 017 −1b)2 0 15 −1 −14 0 2c)2 15 1d)2 0 35 1 10 0 216. If A =2 1 14 −6 0−2 7 2, then Ax = 0 hasa) only one solutionb) two solutionsc) infinitely many solutiond) No solution.MATH 220 MIDTERM EXAMINATION PAGE 1017. Suppose A is a 4 × 5 matrix such that each r ow contains a pivot. Which of the followingstatements is false?a) The columns of A span R4.b) Ax = 0 has a free variablec) The columns of A are linearly independentd) The columns of A are linearly dependent18. Let T : R2→ R2be a linear transformation. If T (e1) =12and T ( 2 e1+ e2) =35, what isthe standard matrix of T ?a)1 32 5b)1 12 1c)1 −12 0d)3 −12 4MATH 220 MIDTERM EXAMINATION PAGE 1119. Let A =1 0 4−2 1 −61 −3 −2and b =b1b2b3. Then Ax = b is consistent ifa) b1+ b2+ b3= 0b) 2b1− b2+ b3= 0c) 5b1+ 3b2+ b3= 0d) b1− b2+ b3= 020. 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 ontoc) If A is a 2 × 3 then the linear transformation defined by T (x) = Ax cannot be o ntod) Two nonzero vectors in Rnare linearly dep endent if one of the vectors is a scalarmultiple of the ot