MATH 251 MATH 251: Multivariate Calculus MATH 251 SPRING 2010 EXAM-I SPRING 2010 EXAM-I EXAMINATION COVER PAGE Professor Moseley

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1 MATH 51 MATH 51: Multivariate Calculus MATH 51 SPRING 010 EXAM-I SPRING 010 EXAM-I EXAMINATION COVER PAGE Professor Moseley PRINT NAME ( ) Last Name, First Name MI (What you wish to be called) ID # EXAM DATE Friday, Feb. 5, 010, 13:30 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. 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, you should eplain your solutions fully and carefully. For regrades, your entire solution will 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 may be given on regrades 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.) Date Signature Total 1

2 MATH 51 EXAM 1 Spring 010 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=, A =, and B =. Compute the following: i 1. ( pt.) A =. A B C D E. ( pt.) A T =. A B C D E. 3. ( pt.) A* =. A B C D E 4. ( pt.) αa =. A B C D E 5. ( pts.) A+B =. A B C D E 6. (4 pts.) AB =. A B C D E Possible answers this page. 1 i 1 i 0 1 i 1 i 1 i i 1 1+ i 1- i i 1 + i i 1 i 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 i 1 + i 1 1 i 1 i i 0 i AD) AE) BC) BD) BE) CD) CE) i i 0 i + i 0 i i 0 i + i 0 DE) ABC) ABD) ABE) ACD) 1+ i 3 i i 1 ACE) ABCDE)None of the above. Possible points this page = 14. POINTS EARNED THIS PAGE 1 i 0 1 i 1 1 i i 1 1 i 1+ i 3 i 0 1 i

3 MATH 51 EXAM 1 Spring 010 Prof. Moseley Page 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. ( pts.) If α is a scalar and C = αa, then c ij =. A B C D E 8. ( 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) a ij B) βa ij C) αa i D) b ij a ij E) b ij + a ij AB) a ij /b ij AC) a b AD) n AE) a b BC) a ij +c ij BD) b ij BE) b ij d ij CD) b ij + e ij CE) a ij b ij ABCDE)None of the above j1 ij ij n i1 ij ij n k 1 a b ik kj 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 not commutative. 11. (1 pt.) A)True or B)False α,βr and AR m n, α(βa) is equal to (αβ)a. 1. (1 pt.) A)True or B)False Multiplication of square matrices is not associative. 13. (1 pt.) A)True or B)False Multiplication of square matrices is commutative. 14. (1 pt.) A)True or B)False There eists a 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 but (AB) -1 is not equal to A -1 B (1 pt.) A)True or B)False If A is an invertible square matri, then (A -1 ) -1 eists and (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 =

4 MATH 51 EXAM 1 Spring 010 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 = 1 possibilities below = 3 Recall that this scalar system can be written as the vector equation A=b where = [ 1,, 3, 4 ] T. Be careful as a single mistake may make all of your answers wrong and there is no part credit. 0.(3 pts.) 1 =. A B C D E 1.(3 pts.) =. A B C D E.(3 pts.) 3 =. A B C D E 3.(3 pts.) 4 =. A B C D E Possible answers this page. A) 0 B) 1 C) D) 3 E) 4 AB) 5 AC) 6 AD) 7 AE) 8 BC) 9 BD) 10 BE) 1 CD) CE) 3 DE) 4 ABC) 5 ABD) 6 ABE) 7 BCD) 8 BCE) 9 BDE) 10 CDE)None of the numbers listed above is correct as A=b has no solution. ABCD) None of the numbers listed is correct as A=b does not have a unique solution. ABCDE) A=b has a unique solution, but none of the above answers is correct for this component of. Possible points this page = 1. POINTS EARNED THIS PAGE =

5 MATH 51 EXAM 1 Spring 010 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 0 4. (1 pt.) A)True or B)False If, then (*) has at least one solution. 5. (1 pt.) A)True or B)False The vector equation (*) always has at least one solution. 6. (1 pt.) A)True or B)False If A is square (n=m) and nonsingular, then (*) always has a unique solution. 7. (1 pt.) A) True or B)False The equation (*) can be considered as a mapping problem from one vector space to another. 1 i 8. (1 pt.) A)True or B)False If A then (*) has a unique solution. i 1 Total points this page = 5. TOTAL POINTS EARNED THIS PAGE

6 MATH 51 EXAM 1 Spring 010 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 i A b A b Prob(C, ); 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 9 (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 i 1 i 1 1 i 1 1 i i C) D) E) AB) AC)None of the above A b 30. ( 4 pts.) The solution of may be written as 1. A B C D E A) No Solution B) 0 1 i 1 i 1 i 1 i 1 i C) D) E) AB) AC) AD) 0 y 1 y 0 1 y 0 1 y 0 1 y 0 1 BC) None of the above correctly describes the solution or collection of solutions. Total points this page = 8. TOTAL POINTS EARNED THIS PAGE

7 MATH 51 EXAM 1A- Spring 010 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 31. (1 pt.) A)True or B)False + ( + ) = ( + ) + is not one of the eight aiomatic properties in the definition of a vector space. y y 3. (1 pt.) A)True or B)False + = + is not 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 one of the eight aiomatic properties in the definition of a vector space. 34. (1 pt.) A)True or B)False For each V, there eist a vector, denoted by, such that + ( ) = 0 is one of the eight aiomatic properties in the definition of a vector space 35. (1 pt.) A)True or B)False α (ß ) = ( αß) is one of the eight aiomatic properties in the definition of a vector space Total points this page = 5. TOTAL POINTS EARNED THIS PAGE

8 MATH 51 EXAM 1 Spring 010 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 36. (1 pt.) A)True or B)False S is a spanning set for W if W, c 1, c,..., c k in K such that c c. 1 1 k k 37. (1 pt.) A)True or B)False S is a linearly independent set if the only solution to c c 0 is c 1 = c = = c k = k k 38. (1 pt.) A)True or B)False S is not a basis for W if it is linearly dependent and spans W. 39. (1 pt.) A)True or B)False A basis set for W is unique. 40. (1 pt.) A)True or B)False The dimension of R n is not n. Total points this page = 5. TOTAL POINTS EARNED THIS PAGE

9 MATH 51 EXAM 1 Spring 010 Prof. Moseley Page 8 Follow the instructions on the Eam Cover Sheet for Fill-in-the Blank/Multiple Choice questions. 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. 41. (4 pts.) Let S = { v }R 3 where = [,, 6] T and = [3, 3,9] T 1, v v 1 v. Then S is. A B C D E A) S is linearly independent as c 1 v 1 + c v = [0,0,0] implies c 1 = 0 and c = 0. B) S is linearly independent as 3 v 1 + () v = [0,0,0]. C) S is linearly dependent as c 1 v 1 + c v = [0,0,0] implies c 1 = 0 and c = 0. D) S is linearly dependent as 3 v 1 + () v = [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. 4. (4 pts.) Let S = { v }R 3 where = [, 4, 8] T and = [3, 6, 11] T 1, v v 1 v. Then S is. A B C D E A) S is linearly independent as c 1 v 1 + c v = [0,0,0] implies c 1 = 0 and c = 0. B) S is linearly independent as 3 v 1 + () v = [0,0,0]. C) S is linearly dependent as c 1 v 1 + c v = [0,0,0] implies c 1 = 0 and c = 0. D) S is linearly dependent as 3 v 1 + () v = [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. Total points this page = 8. TOTAL POINTS EARNED THIS PAGE

10 MATH 51 EXAM 1 Spring 010 Prof. Moseley Page Let A = Use the back of the previous sheet to compute the determinant of A. Write your answer in the blank and then circle your answer to each of the following questions. 43. ( 3 pts.) The first step of the Laplace Epansion in terms of the first column yields det(a) = A B C D E A) (1) 1 0 (1) 0 4 B) (1) 1 0 (3) 0 4 C) C) (3) 1 0 ( 3) 0 4 D) (1) 1 4 (3) 0 4 E) AB) (3) 1 4 (3) 1 0 AC (1) 1 4 (3) 1 0 AD) ABCDE) None of the above. 44. (3 pts.) The first step in using Gauss Elimination to find det(a) yields (1) 1 0 (3) (1) 1 4 (3) (1) 1 4 (3) det(a) =. A B C D E A) B) C) D) E) AB) AC) AD) AE) BC) BD) BE) CD) ABCDE) None of the above (4 pts.) The numerical value of det(a) is det(a) =. A B C D E A) 1 B) C) 3 D) 4 E) 5 AB) 0 AC) 1 AD) AE) 3 BC) 4 BE)5 CD) None of the above. Possible points this page = 10. POINTS EARNED THIS PAGE =

11 MATH 51 EXAM 1 Spring 010 Prof. Moseley Page 10 PRINT NAME ( ) ID No. Follow the instructions on the Eam Cover Sheet for Fill-in-the Blank/Multiple Choice questions. Also circle your answer. Let a and b be the vectors, a = <,1,> = (,1,) = [,1,] T = î ĵ + ˆk and b = <1,,1> = (1,,1) = [1,,1] T = î ĵ ˆk. 46. (3 pts.) Then the dot product is a b = ( a, b ) = a, b =. A B C D E 47. (5 pts.) The cross product is a b =. A B C D E Possible answers this page. A) 1 B) C) 3 D) 4 E) 5 AB) 1 AC) AD) 3 AE) 4 BC) iˆ 6j ˆ kˆ BD) BE) i ˆ 6j ˆ k ˆ CD) 3i ˆ 3j ˆ kˆ CE) 4i ˆ j ˆ kˆ DE) 5i ˆ ˆj kˆ ABC) 6i ˆ kˆ ABD) 3i ˆ j ˆ kˆ ABE) 3i ˆ j ˆ 3k ˆ BCD) 3i ˆ j ˆ kˆ BCE) 3i ˆ j ˆ kˆ CDE) 3i ˆ j ˆ kˆ ABCD) None of the above. iˆ 6j ˆ kˆ

12 Possible points this page = 8. POINTS EARNED THIS PAGE = MATH 51 EXAM 1 Spring 010 Prof. Moseley Page 11 PRINT NAME ( ) ID No. Follow the instructions on the Eam Cover Sheet for Fill-in-the Blank/Multiple Choice questions. a + b y = 3 c + d y = 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. 48. (4 pts.) =. A B C D E 49.(4 pts.) y =. A B C D E Possible answers. 3 b b d 3 d A) B) C)) D) E) AB) AC) AD) ad bc ad bc a a c c b b a 3 a a 3 a d ad bc 3 d ad bc c ad bc c 3 ad bc c ad bc c 3 ad bc AE) BC) BD) BE) CD) DE) ABC) ABD) ad bc ad bc a b b 3 b 3 3 b b 3 3 a c d ad bc d ad bc d bc ad d bc ad d bc ad c bc ad

13 a c ABE) ACD) ACE) ABCDE) None of the above bc ad b d bc ad a b bc ad Possible points this page = 8. POINTS EARNED THIS PAGE =

MATH 251 MATH 251: Multivariate Calculus MATH 251 FALL 2009 EXAM-I FALL 2009 EXAM-I EXAMINATION COVER PAGE Professor Moseley

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