MATH 220 NAMEFINAL EXAMINATION STUDENT NUMBERMAY 2, 2006 INSTRUCTORFORM A SECTION NUMBERThis examination will be machine processed by the University Testing Service. Use onlya number 2 pencil on your answer sheet. On your answer sheet identify your name, thiscourse (Math 220) and the date. Code and blacken the corresponding circles on your answersheet for your student I.D. number and class section number. Code in your test f orm. FIVEPOINTS WILL BE DEDUCTED FROM YOUR FINAL SCORE IF YOU DO NOT FILLIN YOUR ID NUMBER, SECTION NUMBER OR TEST VERSION CORRECTLY.There are 25 multiple choice questions each worth six points. For each problem four possibleanswers are given, only one of which is correct. You should solve the pro blem, no te the letterof the answer that you wish to give and blacken the corresponding space o n the answersheet. Mark only one choice; darken the circle completely (you should not be able to see theletter after you have darkened the circle). Check frequently t o be sure the problem numberon the test sheet is the same as the problem number of the answer sheet.THE USE OF A CALCULATOR, CELL PHONE, OR ANY OTHER ELECTRONIC DE-VICE IS NOT PERMITTED DURING THIS EXAMINATION.CHECK THE EXAMINATION BOOKLET BEFORE YOU START. THERE SHOULD BE25 PROBLEMS ON 14 PAGES (INCLUDING THIS ONE).MATH 220 FINAL EXAMINATION, FORM A PAGE 21. Which of the following is the coefficient matrix for a homogeneous system Ax = 0 with onlythe trivial (zero) solution?a) A =1 0 0 −20 1 0 00 0 0 3b) A =1 0 0 00 1 0 00 0 1 0c) A =1 2 30 0 10 0 0d) A =1 2 30 4 50 0 62. Find the condition that b1, b2, b3must satisfy in order for the linear systemx1+ x2+ 2x3= b1x1+ 2x2+ 3x3= b23x1+ 4x2+ 7x3= b3to be consistent.a) b3= −2b1+ 5b2.b) b3= 3b2.c) b3= 2b1+ b2.d) The system is consistent for any b1, b2, b3.MATH 220 FINAL EXAMINATION, FORM A PAGE 33. Suppose {u, v, w} is a linearly independent set of vectors in R3. Which of the following setsis also linearly independent?a) {u, u − v, u − w}b) {0, v}c) {u, u + v, u + w, v −w}d) {u, u − v, v}4. Let A =1 0 32 −1 00 2 −2, then the span of the columns of A is:a) {0}.b) a line.c) a plane.d) all of R3.MATH 220 FINAL EXAMINATION, FORM A PAGE 45. Suppose S : R2−→ R2is given by S(x1, x2) = (0, 0) and T : R3−→ R3is given byT (x1, x2, x3) = (x1, 2x2, x3+ 3), then which of the following is t rue ?a) Neither S nor T is a linear map.b) S is a linear map but T is not.c) S is not a linear map, but T is.d) Both S and T and linear maps.6. Let T : R2−→ R2be a linear map. Suppose t hat T2−1=34and T12=−21.What is T3−4?a)15b)87c)21d)178MATH 220 FINAL EXAMINATION, FORM A PAGE 57. Find the standard matrix of the linear map T : R2−→ R2which first expands in the x1-direction by a factor of two and in the x2-direction by a factor of three, then reflects acrossthe x2-axis.a)−3 10 −2b)3 00 −2c)2 −10 3d)−2 00 38. Suppose A =1 30 1, B =−1 02 2, C =2 −31 0. Find A2− BC + 2AT.a)5 30 9b)1 10 9c)5 6−6 9d)11 127 3MATH 220 FINAL EXAMINATION, FORM A PAGE 69. Given that A =4 −2 05 3 2−1 1 0is an invertible matrix. What is the second column of A−1?a)010b)1/3−11c)1/400d)001/210. Let A be an n × n matrix and suppose the equation Ax = b is inconsistent, then which ofthe following statements is true?a) b is in the column space o f A.b) A is row equivlant to In.c) A has less than n pivot columns.d) ATis invertible.MATH 220 FINAL EXAMINATION, FORM A PAGE 711. Which of the following is a subspace o f R3?a)x1x2x3: x1< x2.b) The x1, x2- plane.c) The point210.d) The set of vectors of t he form010+ t100for any scalar t.12. Let A be an m × n matrix. Which of the following is always true?a) Rank A = m.b) The column space of A is a subspace of Rn.c) The null space of A has dimension n.d) Rank A+dim Nul A = nMATH 220 FINAL EXAMINATION, FORM A PAGE 813. Let A =5 2 0−1 3 14 −2 0, then what is det A?a) −18b) 18c) 12d) −1214. Let A =−4 0 51 2 17 6 −3, and B =7 6 −31 2 1−4 0 5. Which of the following is true?a) det A = 0.b) det B = 2.c) det A =det B.d) det A = −det B.MATH 220 FINAL EXAMINATION, FORM A PAGE 915. Find the least squares solutionˆx of Ax = b where A =1 01 31 6and b =145.a)ˆx =42b)ˆx =21c)ˆx =11d)ˆx =4/32/316. Let A be an n × n matrix. Which of the f ollowing is NOT always true?a) The characteristic equation of A is of the form p(λ) = 0, where p is a degree npolynomial.b) If A is similar to B then A and B have the same eigenvalues.c) A has at most n distinct eigenvalues.d) If 1 is an eigenvalue of A, then A is invertible.MATH 220 FINAL EXAMINATION, FORM A PAGE 1017. Let A =3 21 11 00 21 −2−1 3, noting that3 21 1−1=1 −2−1 3. What is A3?a)5 17−4 22b)32 015 2c)−13 42−7 22d)16 −32−8 1618. Let B = {b1, b2, b3} be a basis f or R3, and let T : R3−→ R3be a linear map such t hat[T ]B=2 1 31 −1 00 1 −2. What is T (b1+ b2− b3)?a) 3b3b) b1+ b2− b3c) 3b1+ b2+ b3d) 2b1MATH 220 FINAL EXAMINATION, FORM A PAGE 1119. Let A be a 3 ×3 matrix with real number entries and suppose that λ1= 3 and λ2= 1 + 2iare two of its eigenvalues. What is the third (real or complex) eigenvalue of A?a) It cannot be determined from the info r matio n givenb) −3c) −1 + 2id) 1 − 2i20. What is the distance between u =1−123and v =1234?a)√11b)√5c) 9d) 5MATH 220 FINAL EXAMINATION, FORM A PAGE 1221. Which of the following statements is NOT always true?a) If U is an orthogonal matrix, t hen U is invertible.b) If U is an orthogonal matrix, then det U > 0.c) The columns of an orthogonal matrix form an orthonormal set.d) If U is an n ×n orthogonal matrix, then kUxk = kxk for all vectors x in Rn.22. Given that B = {u1, u2, u3} is an ort hogonal set, where u1=3−30, u2=22−1, u3=114,what is [x]Bfor x =246?a)−1/32/35/3b)2/34/32c)−6630d) None of the aboveMATH 220 FINAL EXAMINATION, FORM A PAGE 1323. Let W be