MATH FINAL EXAM DECEMBER 8, 7 FORM A NAME STUDENT NUMBER INSTRUCTOR SECTION NUMBER This examination will be machine processed by the University Testing Service. Use only a number pencil on your answer sheet. On your answer sheet, identify your name, this course (MATH ) and the date. Code and blacken the corresponding circles on your answer sheet for your student I.D. number and the class section number. Code in your test form. There are 5 multiple choice questions each worth six points. For each problem, four possible answers are given, only one of which is correct. You should solve the problem, note the letter of the answer that you wish to give and blacken the corresponding space on the answer sheet. Mark only one choice; darken the circle completely (you should not be able to see the letter after you have darkened the circle). Check frequently to be sure the problem number on the test sheet is the same as the problem number of the answer sheet. THE USE OF A CALCULATOR, CELL PHONE, OR ANY OTHER ELEC- TRONIC DEVICE IS NOT PERMITTED DURING THIS EXAMINATION. CHECK THE EXAMINATION BOOKLET BEFORE YOU START. THERE SHOULD BE 5 PROBLEMS ON 4 PAGES (INCLUDING THIS ONE).
MATH FINAL EXAM, FORM A PAGE. Find all solutions of the following system of linear equations: a) x = 5, x =, x 3 =. b) x = 5 + x 3, x =, x 3 is free. c) x = + x 3, x =, x 3 is free. d) There are no solutions. x + x x 3 = x + 5x x 3 = 5x + x 5x 3 =. Suppose 3 5 is the augmented matrix of a linear system. For which 5 5 h 3 value(s) of h is this system consistent? a) h = b) h c) h = d) h
MATH FINAL EXAM, FORM A PAGE 3 3. If A = 3 6 8 4 has a solution. a) b 3 = b + b b) b 3 = b + b c) b 3 = 3b + 3b d) For any b, b, b 3., and b = b b b 3, then describe the set of all b such that Ax = b 4. If A = 3 5 5, then what is the middle row of the inverse matrix A? 4 a) [ ] b) [ 6 ] c) [ ] d) A does not exist.
MATH FINAL EXAM, FORM A PAGE 4 5. If A =, and the linear transformation is given by T(x) = Ax, which of the 4 following vectors is in the range of T? a) [ ] b) [ ] c) d) [ 6. If T( 3 a) [ b) [ 8 c) [ 9 d) ] ] ] ] [ ) = 3 ] [, and T( ] [ ) = ] [ 4, then what is T( 5 ] ) =?
MATH FINAL EXAM, FORM A PAGE 5 7. What is a parametric form of the solutions to the following linear system? a) x = t, x = t, x 3 = t b) x = t + s, x = s, x 3 = t c) x = t, x = t, x 3 = d) x = t, x = t, x 3 = t x + x x 3 = 3x 4x + 7x 3 = x 3x + 5x 3 = 8. Let the vectors v, v and v 3 be given by v =, v = 5, and v 3 =. 5 5 The basis B = {v,v,v 3 } is not orthogonal. What is the value of x if the vector y = is given by the linear combination y = v x + v x + v 3 x 3. a) - b) 5 c) d) -3
MATH FINAL EXAM, FORM A PAGE 6 9. Let the vectors v, v and v 3 be given by v = / /, v = / /, and v 3 =. The basis B = {v,v,v 3 } is orthonormal. What is the value of c if the vector y = is given by the linear combination a) b) c) / d) y = v c + v c + v 3 c 3.. T : R R first reflects points through the horizontal (x ) axis, and then rotates clockwise by 9. Find the standard matrix of T. a) b) c) d)
MATH FINAL EXAM, FORM A PAGE 7. Suppose A is an n n matrix given by A = [ a... a n ], and the linear transformation x Ax is onto R n. Which of the following are true? I - c a +... + c n a n = a n for some c,...,c n in R. II - det A III - The solution set of Ax = is a two-dimensional subspace of R n. IV - The linear transformation x A x is onto R n. a) I and IV b) II and IV c) II and III d) I and III 3 3. The matrix A given by A = 4 4 6. is row equivalent to the matrix 3 3 7/ 4 3/4 /. Which of the following is a basis for the null space of A? a) 3, 4 3 8 b) c) 7/ 3/4, 4 / 4 7/ d) /, 3/4
MATH FINAL EXAM, FORM A PAGE 8 3. Let A = 4 4 8. Any solution of the equation Av = may be written as 3 v = t + s, where t, s R. What is the rank of the matrix A? a) b) c) 3 d) 4 4. What is det B if B is given by B = 3 3? a) b) 6 c) d) 4
MATH FINAL EXAM, FORM A PAGE 9 5. Suppose the linear transformation T : R 3 R 3 is given by x Ax. Further, suppose T reflects any vector in R 3 across a plane through the origin. What is the dimension of the eigenspace associated with the eigenvalue λ =? a) b) c) d) 3 ( (3A T 6. Suppose det A = a and det B = b. What is the value of det ) ) B. a) 9a b b) 3a /b c) 9a /b d) 3a b
MATH FINAL EXAM, FORM A PAGE 5/ / 7. Let the matrix A be given by / 5/. Which of the following is an eigenvector of A? a) b) c) d) 8. Let the matrix A, given by A = as A = P DP, where 5 D =, P = What is the matrix product PA 6? a) 6 5 6 5 b) 6 6 6 5 c) 6 6 5 6 + 6 5 6 6 5 6 + 6 5 6 d) 6 5 6 6, be diagonalizable. That is, A may be written 5 5, and P =.
MATH FINAL EXAM, FORM A PAGE 9. Find the characteristic polynomial of A = a) ( λ)( λ)(3 λ) b) ( λ)( λ)( λ) c) ( λ)(λ λ 3) d) ( λ)(λ + λ + ) 3.. Suppose the matrix A is given by. Which of the following is an eigenvalue/eigenvector pair? (Note that in the choices below, i =.) a) λ = + i i, b) λ = + i i +, i i c) λ = + i, i + d) λ = + i, i +
MATH FINAL EXAM, FORM A PAGE. What is the distance from y = a) 4 b) 3 c) d) 3 to the line through u =?. For which value of h are the vectors u = a) h = b) h = c) h = 3 d) h = 4 and u = h orthogonal?
MATH FINAL EXAM, FORM A PAGE 3 3. Which one of the following bases is orthogonal? / 8 a) 4/ 8 /, / 8, /3 /3 /3 b),, c),, / 3 d) / 3 /, / 3 /, 4. Find the distance from the point 5 5 to the plane spanned by and. a) b) c) 5 d) 5
MATH FINAL EXAM, FORM A PAGE 4 5. Construct an orthogonal basis for Span{x,x }, where x = a) 5 and 4 5 b) and 4 5 c) and 3 3 d) and 4 and x = 4 5 6. Let the matrix A be given by A = 5 5, What is the dimension of the eigenspace 4 corresponding to the eigenvalue λ = 4? a) b) c) d) 3