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PSU MATH 220 - STUDY NOTES MATH 220

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MATH 220 FINAL EXAMINATION DECEMBER 16, 2004Name ID # Section #There are 25 multiple choice questions. 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 answerthat you wish to give and blacken the corresponding space on the answer sheet. Mark onlyone choice; darken the circle completely (you should not be able to see the letter afteryou 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 25 PROBLEMSON 14 PAGES (INCLUDING THIS ONE).MATH 220 FINAL EXAMINATION PAGE 21. Consider the matrices A =1 2 0 50 0 1 30 0 0 1and B =1 2 0 5 00 0 1 3 00 0 0 0 1. Which of the followingstatements is t rue?a) None of the matrices is in r educed echelo n formb) Both A and B are in reduced echelon formc) Only A is in reduced echelon formd) Only B is in reduced echelon form2. What is the general solution of the linear system2x1+ x2+ x3= 10x2− x3= 4x1− 2x2+ 3x3= −5?a)x1= 3 − 2x3x2= 4 + x3x3is freeb)x1= 1 − 2x2x3= 4 − x2x2is freec)x1= 3 − x3x2= 4 + x3x3is freed) None of the aboveMATH 220 FINAL EXAMINATION PAGE 33. If T is a linear transformation whose standard matrix is given by A =3 21 05 1, then whichof 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 a nd onto.d) T is neither one-to-one nor onto.4. Let A =1 2 50 −2 −6−1 3 10and b =b1b2b3. Then Ax = b is consistent ifa) 2b1+ 5b2+ 2b3= 0b) 2b1− b2+ 5b3= 0c) b1− 3b2+ b3= 0d) b1+ b2+ b3= 0MATH 220 FINAL EXAMINATION PAGE 45. Find the parametric vector form of the solution set of the systemx1+ x2− 2x3= 52x1+ 3x2+ 4x3= 2a) x = x311−2b) x =13−80+ x310−81c) x = x213−80+ x311−2d) None of the above6. Let v1=111, v2=−141, v3=112. Which of the following statements is true?a) Span{v1, v2, v3} is the origin.b) Span{v1, v2, v3} is a line through the origin.c) Span{v1, v2, v3} is a plane through the origin.d) Span{v1, v2, v3} is all of R3.MATH 220 FINAL EXAMINATION PAGE 57. Consider the two planes given by 2x1+ 3x2− 5x3= 7 a nd x1− x2+ 5x3= 1. Which o f thefollowing statements is true?a) Their intersection is empty.b) Their intersection is the point (2, 1, 0).c) Their intersection is a line through the point (−2, 3, 1) with direction u =210.d) None of the above statements describes the intersection of these planes.8. Let A be the standard matrix of a linear transformation T : Rn→ Rnthat is onto. Whichof the following statements is true?a) A is not invertible.b) Ax = 0 has a only the trivial solution.c) The columns of A are linearly dependent.d) T is not one-to-one.MATH 220 FINAL EXAMINATION PAGE 69. Suppose T is the linear transformation that first rotates points throughπ4radians counter-clockwise and then projects points onto the x-axis. The standard matrix of T isa)√22−√220 0b)−√22√220 0c)0 0−√22√22d) None of the above10. Which of the following sets is a subspace of R3?a)x1x2x3| x1+ x2+ x3= 7b)x1x2x3| x2+ x3= 0c)x1x2x3| sin(x2) − x3= 0d)x1x2x3| x21+ x22= 0MATH 220 FINAL EXAMINATION PAGE 711. L et A =1 2 3 41 1 4 2−1 −2 5 3. What is the dimension of the null-space of A?a) 1b) 2c) 3d) 412. L et A =2 −3 0 21 0 3 10 2 4 0. Find a basis for the column space of A.a)210,b)210,−302c)210,−302,034d) None of the aboveMATH 220 FINAL EXAMINATION PAGE 813. L et A =1 1 10 1 2−1 5 11. Which of the following vectors is in the null-space of A?a)01−1b)−110c)121d)1−2114. What is the characteristic polynomial of A =1 2 30 1 20 2 1?a) (1 − λ)(1 + λ)(λ − 3)b) (1 − λ)2(λ + 1)c) (1 − λ)(λ + 2)d) None of the aboveMATH 220 FINAL EXAMINATION PAGE 915. The determinant of A =0 2 0 0−3 0 1 00 0 0 40 0 1 4isa) 24b) −24c) 0d) None the above.16. L et A and B be 3 × 3 matrices such that det(A) = 4 and det(B) = 3. Find det(2A−1B2).a)92b) 72c) −18d) 18MATH 220 FINAL EXAMINATION PAGE 1017. Find the eigenvalues of A =1 5 30 3 00 2 2.a) λ = 1, 2, 3b) λ = 0, 1, 2c) λ = 1, 3, 5d) λ = 2, 3, 518. L et T : R2→ R2be a linear transformation whose standard matrix is A =−2 11 −2. Finda basis B such that the B-matrix of T is diagonal.a) B =10,−11b) B =01,−11c) B =11,21d) B =11,−11MATH 220 FINAL EXAMINATION PAGE 1119. Which of the following sets is an orthonormal set?a)111,−101b)1/√32/√32/√3−1/√3c)1/√31/√31/√3,−2/√61/√61/√6d)111,00020. If A =2 11 1−1 00 31 −1−1 2. Then A3equalsa)−29 56−28 55b)−24 56−25 55c)−29 56−28 57d) None of the aboveMATH 220 FINAL EXAMINATION PAGE 1221. L et B =21−1,011,1−11. Find the coordinate vector o f x =123relative to theorthogonal basis B for R3a)1/65/22/3b)1/65/22c)1/65/22/3d) None of the above22. What is the distance between x =126and the line L through the origin that is spannedby u =111?a)√14b)√26c)√6d) None of the aboveMATH 220 FINAL EXAMINATION PAGE 1323. A diagonalization of A =3 11 3isa)−1 11 12 00 4−1 11 1b)−1 11 12 00 4−1/2 1/21/2 1/2c)−1 11 11 00 81 −1−1 −1d) None of the above24. Which of the following statements is false?a) An n × n matrix with n distinct real eigenvalues is diagonalizable.b) If A is invertible then A is diagonalizable.c) Similar matrices have the same determinant .d) Similar


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