MATH 251 MATH 251: Multivariate Calculus MATH 251 FALL 2009 EXAM-I FALL 2009 EXAM-I EXAMINATION COVER PAGE Professor Moseley
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1 MATH 251 MATH 251: Multivariate Calculus MATH 251 FALL 2009 EXAM-I FALL 2009 EXAM-I EXAMINATION COVER PAGE Professor Moseley PRINT NAME ( ) Last Name, First Name MI (What you wish to be called) ID # EXAM DATE Friday, February 6, 2009, 11:30 I swear and/or affirm that all of the work presented on this eam is my own and that I have neither given nor received any help during the eam. Date Signature SIGNATURE DATE INSTRUCTIONS: Besides this cover page, there are 11 pages of questions and problems on this eam. MAKE SURE YOU HAVE ALL THE PAGES. If a page is missing, you will receive a grade of zero for that page. Read through the entire eam. If you cannot read anything, raise your hand and I will come to you. Place your I.D. on your desk during the eam. Your I.D., this eam, and a straight edge are all that you may have on your desk during the eam. NO CALCULATORS! NO SCRATCH PAPER! Use the back of the eam sheets if necessary. You may remove the staple if you wish. Print your name on all sheets. Pages 1-11 are Fillin-the Blank/Multiple Choice or True/False. Epect no part credit on these pages. For each Fill-in-the Blank/Multiple Choice question write your answer in the blank provided. Net find your answer from the list given and write the corresponding letter or letters for your answer in the blank provided. Then circle this letter or letters. There are no free response pages. However, to insure credit, you should eplain your solutions fully and carefully. Your entire solution may be graded, not just your final answer. SHOW YOUR WORK! Every thought you have should be epressed in your best mathematics on this paper. Partial credit will be given as deemed appropriate. Proofread your solutions and check your computations as time allows. GOOD LUCK!! REQUEST FOR REGRADE Please regrade the following problems for the reasons I have indicated: (e.g., I do not understand what I did wrong on page.) Scores page points score (Regrades should be requested within a week of the date the eam is returned. Attach additional sheets as necessary to eplain your reasons.) I swear and/or affirm that upon the return of this eam I have written nothing on this eam ecept on this REGRADE FORM. (Writing or changing anything is considered to be cheating.)
2 23 Total 100 MATH 251 EXAM 1A-1 Fall 2009 Prof. Moseley Page 1 Follow the instructions on the Eam Cover Sheet for Fill-in-the Blank/Multiple Choice questions. For each question write your answer in the blank provided. Net find your answer from the list of possible answers listed below and write the corresponding letter or letters for your answer in the blank provided. Finally, circle this letter or letters. 1+ i 1 i i 1 Let α= 2, A =, and B =. Compute the following: i 1. (2 pt.) A =. A B C D E 2. (2 pt.) A T =. A B C D E. 3. (2 pt.) A* =. A B C D E 4. (2 pt.) αa =. A B C D E 5. (2 pts.) A+B =. A B C D E 6. (4 pts.) AB =. A B C D E Possible answers this page. 1 i 1i 0 1i 1 i 1 i i 1 1+i 1-i i 1+i i 1i i 1+i 1 0 A)) B) C) D) E) AB) AC) 1 i 1 1 i 0 1 i 1 1+ i 0 1 i 0 1+ i i 0 2 i i 1 1 i 1 2i 2i 0 2i AD) AE) BC) BD) BE) CD) CE) 1 i 0 1 i 1 1 2i 2 i 1 1 i
3 2 2i 22i i 2 + 2i i 2 2i i 2 + 2i 0 2 DE) ABC) ABD) ABE) ACD) 1 + i 3 i i 1 ACE) ABCDE)None of the above. 1 + i 3 i 0 1 i Possible points this page = 14. POINTS EARNED THIS PAGE MATH 251 EXAM I Fall 2009 Prof. Moseley Page 2 For questions 7, 8, and 9, follow the instructions on the Eam Cover Sheet for Fill-in-the Blank/Multiple Choice questions. For each question write your answer in the blank provided. Net find your answer from the list. of possible answers listed below and write the corresponding letter or letters for your answer in the blank provided. Finally, circle this letter or letters. Using the abbreviated (tensor) notation for a matri discussed in class, let A = [a ij ], B=[b ij ], C=[c ij ], D=[d ij ], and E=[e ij ] be square nn matrices. 7. (2 pts.) If α is a scalar and C = αa, then c ij =. A B C D E 8. (2 pts.) If D = A + B, then d ij =. A B C D E 9. (3 pts.) If E = AB, then e ij =. A B C D E Possible Answers for questions 7, 8, and 9. A) βa ij B) αa i C) b ij a ij D) b ij + a ij E) a ij /b ij AB) a b AC) a b AD) AE) a ij BC) a ij +c ij BD) b ij BE) b ij d ij CD) b ij + e ij CE) a ij b ij DE) None of the above n i1 ij ij n k1 ik kj n j1 a b ij ij True or False. Matri Algebra. Circle True or False, but not both. If I cannot read your answer, it is wrong. 10. (1 pt.) A)True or B)False Matri addition is commutative. 11. (1 pt.) A)True or B)False α,βr and AR m n, α(βa) = (αβ)a. 12. (1 pt.) A)True or B)False Multiplication of square matrices is associative. 13. (1 pt.) A)True or B)False Multiplication of square matrices is commutative. 14. (1 pt.) A)True or B)False There eist no matri BR m n such that AR m n, we have A + B = A. 15. (1 pt.) A)True or B)False If A is an invertible square matri, then (A T ) -1 eists and (A T ) -1 = (A -1 ) T. 16. (1 pt.) A)True or B)False If A and B are invertible square matrices, then (AB) -1 eists and (AB) -1 = A -1 B (1 pt.) A)True or B)False If A is an invertible square matri, then (A -1 ) -1 eists and
4 (A -1 ) -1 = A. 18.(1 pt.) A)True or B)False If A and B are square matrices, then (AB) T eists and (AB) T = A T B T. 19. (1 pt.) A)True or B)False If A is a square matri, then (A T ) T eists and (A T ) T = A. Possible points this page = 17. POINTS EARNED THIS PAGE = MATH 251 EXAM I Fall 2009 Prof. Moseley Page 3 Follow the instructions on the Eam Cover Sheet for Fill-in-the Blank/Multiple Choice questions = 1 Use Gauss elimination to solve this system of linear algebraic equations = 0 Circle the letter or letters that correspond to your answer from the = 0 possibilities below = (3 pts.) 1 =. A B C D E 11. (3 pts.) 2 =. A B C D E 12. (3 pts.) 3 =. A B C D E 13. (3 pts.) 4 =. A B C D E
5 Possible answers this page. A) 0 B) 1 C) 2 D) 3 E) 4 AB) 5 AC) 6 AD) 7 AE) 8 BC) 9 BD) 1 BE) 2 CD) 3 CE) 4 DE) 5 ABC) 6 ABD) 7 ABE) 8 BCD) 9 BCE) 18 CDE) None of the above. Possible points this page = 12. POINTS EARNED THIS PAGE = MATH 251 EXAM I Fall 2009 Prof. Moseley Page 4 True or false. Solution of Linear Algebraic Equations having possibly comple coefficients. Assume A is an m n matri of possibly comple numbers, that is an n 1 column vector of b (possibly comple) unknowns, and that is an m 1 (possibly comple valued) column vector. Now consider. (*) A mn n1 b m1 Under these hypotheses, determine which of the following is true and which is false. If true, circle True. If false, circle False. If I can not read your answer, it is wrong. b (1 pt.) A)True or B)False If, then (*) will have eactly one solution. 15. (1 pt.) A)True or B)False The vector equation (*) always has eactly one solution. 16. (1 pt.) A)True or B)False If A is square (n=m) and singular, then (*) always has a unique solution. 17. (1 pt.) A) True or B)False The equation (*) can not be considered as a mapping problem from one vector space to another. 18. (1 pt.) A)True or B)False If A 1 i then (*) has a unique solution. i 1
6 Total points this page = 5. TOTAL POINTS EARNED THIS PAGE MATH 251 EXAM 1 Fall 2009 Prof. Moseley Page 5 Follow the instructions on the Eam Cover Sheet for Fill-in-the Blank/Multiple Choice questions. Also circle your answer. 1 i 1 Let A,, and. Below or on the back of the previous sheet solve i 1 y b 0 A b A b Prob(C 2, ); that is, solve the vector equation. The form of the answer may not be unique. To obtain the answer listed, follow the directions given in class (attendance is mandatory). A b U c 19 (4 pts.) If is reduced to using Gauss elimination we obtain U c =. A B C D E A) 1 i 1 1 i 1 B) i 1 1 i 1 1 i C) D) E) AB) AC)None of the above i A b 20. ( 4 pts.) The solution of may be written as 0. A B C D E A) No Solution B) 1 i i 1 i 1 i 1 i C) D) E) AB) AC) AD) 1 y 1 y 0 1 y 0 1 y 0 1 BC) None of the above correctly describes the solution or collection of solutions. 1 i y 0 1
7 Total points this page = 8. TOTAL POINTS EARNED THIS PAGE MATH 251 EXAM I Fall 2009 Prof. Moseley Page 6 True or false. Definition of a vector space. Recall the following as the beginning of the definition of a vector space. DEFINITION. A nonempty set of objects (vectors), V, together with an algebraic field (of scalars) K, and two algebraic operations (vector addition and scalar multiplication) which satisfy the algebraic properties listed below (Laws of Vector Algebra) comprise a vector space. (Following standard convention, although incorrect, we will sometimes refer to the set of vectors V as the vector space). The set of scalars K are usually either the real numbers R or the comple numbers C in which case we refer to V as a real or comple vector space. Let, y, z ε V be any vectors and α,ß ε K be any scalars. Then the following must hold: The rest of the definition of a vector space consists of the eight aiomatic properties for a vector space. Answer the following true false questions. y z 21. (1 pt.) A)True or B)False + ( + ) = ( + ) + is one of the eight aiomatic properties in the definition of a vector space. y y 22. (1 pt.) A)True or B)False + = + is one of the eight aiomatic properties in the definition of a vector space y 0 z (1 pt.) A)True or B)False There eists a vector such that for every V, + = is not one of the eight aiomatic properties in the definition of a vector space. 24. (1 pt.) A)True or B)False For each V, there eist a vector, denoted by, such that +( ) =0 is not one of the eight aiomatic properties in the definition of a vector space 25. (1 pt.) A)True or B)False α (ß ) = ( αß) is not one of the eight aiomatic properties in the definition of a vector space
8 Total points this page = 5. TOTAL POINTS EARNED THIS PAGE MATH 251 EXAM I Fall 2009 Prof. Moseley Page 7 True or false. Vector Space Theory. Let S { 1,..., k} 0 W V where W is a subspace of a vector space V over the scalars K and for i = 1,...,k. Answer the following true false questions. i 26. (1 pt.) A)True or B)False S is a linearly independent set for W if W, c 1, c 2,..., c k in K such that. 27. (1 pt.) A)True or B)False S is a spanning set if the only solution to is c 1 = c 2 = = c k = (1 pt.) A)True or B)False S is not a basis for W if it is linearly dependent and spans W. 29. (1 pt.) A)True or B)False A basis set for W is not unique. 30. (1 pt.) A)True or B)False The dimension of R n is n.
9 Total points this page = 5. TOTAL POINTS EARNED THIS PAGE MATH 251 EXAM 1 Fall 2009 Prof. Moseley Page 8 Determine Directly Using the Definition (DUD) if the following sets of vectors are linearly independent. As eplained in class, choose the appropriate answer that gives an appropriate method to prove that your choice is correct (attendance is mandatory). Circle your answer. Then write it in the space provided after the word Answer. Finally, circle this letter after the word Answer. Be careful. If you get the concepts of linearly independent and linearly dependent backwards, your grade is zero. 31. (4 pts.) Let S = {[2, 4, 8] T, [3, 6, 12] T }. Circle the correct answer A) S is linearly independent as c 1 [2, 4, 8] T + c 2 [3, 6, 12] T = [0,0,0] implies c 1 = 0 and c 2 = 0. B) S is linearly independent as 3[2, 4, 8] T + (2) [3, 6, 12] T = [0,0,0]. C) S is linearly dependent as c 1 [2, 4, 8] T + c 2 [3, 6, 12] T = [0,0,0] implies c 1 = 0 and c 2 = 0. D) S is linearly dependent as 3[2, 4, 8] T + (2) [3, 6, 12] T = [0,0,0]. E) S is neither linearly independent or linearly dependent as the definition does not apply. AB) None of the above is a correct statement. Answer A B C D E 32. (4 pts.) Let S = {[2, 2, 4] T, [3, 3, 5] T }. Circle the correct answer A) S is linearly independent as c 1 [2,2,4] T + c 2 [3, 3, 5] T = [0,0,0] implies c 1 = 0 and c 2 = 0. B) S is linearly independent as 3[2,2,4] T + (2) [3, 3, 5] T = [0,0,0]. C) S is linearly dependent as c 1 [2,2,4] T + c 2 [3, 3, 5] T = [0,0,0] implies c 1 = 0 and c 2 = 0. D) S is linearly dependent as 3[2,2,4] T + (2) [3, 3, 5] T = [0,0,0]. E) S is neither linearly independent or linearly dependent as the definition does not apply. AB) None of the above is a correct statement. Answer A B C D E
10 Total points this page = 8. TOTAL POINTS EARNED THIS PAGE MATH 251 EXAM 1 Fall 2009 Prof. Moseley Page 10 Follow the instructions on the Eam Cover Sheet for Fill-in-the Blank/Multiple Choice questions. Also circle your answer. Let and be the vectors, = <2,1,1> = (2,1,1) = [2,1,1] T = 2 + and = <1,2,1> = (1,2,1) = [1,2,1] T = (3 pts.) Then the dot product is = (, ) =, =. A B C D E 47. (5 pts.) The cross product is =. A B C D E Possible answers this page. A) 1 B) 2 C) 3 D) 4 E) 5 AB) 1 AC) 2 AD) 3 AE) 4 BC) BD) BE) CD) 3i ˆ 3 ˆ j k ˆ CE) DE) ABC) ABD) ABE) BCD) BCE) CDE) ABCD) None of the above.
11 Possible points this page = 8. POINTS EARNED THIS PAGE = MATH 251 EXAM I Fall 2009 Prof. Moseley Page 11 Follow the instructions on the Eam Cover Sheet for Fill-in-the Blank/Multiple Choice questions. Also circle your answer. a + b y = 2 c + d y = 3 For the system of algebraic equations given above, use Cramer's Rule to obtain and y. Write your answer in the blank and then circle the letter or letters that correspond to your answer from the possibilities listed below. 38. (4 pts.) =. A B C D E 39.(4 pts.) y =. A B C D E Possible answers. A) B) C) D) E) AB) AC) AD) AE) BC) BD) BE) CD) CE) None of the above
12 Possible points this page = 8. POINTS EARNED THIS PAGE =
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