Check that your exam contains 20 multiple-choice questions, numbered sequentially.

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1 MATH 22 MAKEUP EXAMINATION Fall 26 VERSION A NAME STUDENT NUMBER INSTRUCTOR SECTION NUMBER On your scantron, write and bubble your PSU ID, Section Number, and Test Version. Failure to correctly code these items may result in a loss of 5 points on the exam. On your scantron, bubble letters corresponding to your answers on indicated questions. It is a good idea for future review to circle your answers in the test booklet. Check that your exam contains 2 multiple-choice questions, numbered sequentially. Answer Questions 2 on your scantron. Each question is worth 5 points. THE USE OF A CALCULATOR, CELL PHONE, OR ANY OTHER ELECTRONIC DEVICE IS NOT PERMITTED IN THIS EXAMINATION. THE USE OF NOTES OF ANY KIND IS NOT PERMITTED DURING THIS EXAMINATION.

2 MATH 22 MAKEUP EXAMINATION, VERSION A PAGE 2. If x x 2 is the solution of the system of linear equations x what is the value of x? x 2 + x = 5 x + x 2 + 6x = 2 x + 6x 2 + 2x = 5 a) b) c) d) 2 2. How many solutions does the following linear system have? x + x 2 + x = 2 2x 2x 2 + 6x = 9 x 2 x = a) This linear system has no solutions. b) This linear system has a unique solution. c) This linear system has exactly three solutions. d) This linear system has infinitely many solutions.

3 MATH 22 MAKEUP EXAMINATION, VERSION A PAGE. Which of the following statements is true? 7 5 A =, B = 9 2 2, C = 7 8 a) All matrices A, B, and C are in echelon form. b) A and B are in echelon form. c) B is in echelon form and C is in reduced echelon form. d) B and C are in reduced echelon form.. The linear system x x 2 = x x = 2 2x 2 + x = 7 can be rewritten as which augmented matrix below? a) b) c) d) 2 2 7

4 MATH 22 MAKEUP EXAMINATION, VERSION A PAGE 5. If the set S = Span 2,, 5 6 5, which of the following statements is true about S? a) S is a point in R. b) S is a line in R. c) S is a plane in R. d) S is the whole space R. 6. Find the vector u which satisfies the equation [ ] u = u. a) u = b) u = c) u = d) u = 9 9

5 MATH 22 MAKEUP EXAMINATION, VERSION A PAGE 5 7. Find all the solutions of the given augmented matrix a) x 5 2 b) 5 + x 6 8 c) 6 8 d) 2 or Describe all solutions of Ax = b, where A = 2 and b =. 5 a) 2 + x 2 b) x 2 c) 2 d) There are no solutions.

6 MATH 22 MAKEUP EXAMINATION, VERSION A PAGE 6 9. If A is a 5 7 matrix, which of the following statements about A is false? a) The homogeneous equation Ax = has infinitely many solutions. b) The columns of A are linearly dependent. c) Ax = b will have a unique solution for every b in R 5. d) A has at most 5 pivots.. Consider the matrix equation Ax = b with 5 A = 2 7. Which of the following statements is true? a) The system is always inconsistent for all b. b) The system is consistent for some b. c) The system has a unique solution for all b. d) The system has infinitely many solutions for all b.

7 MATH 22 MAKEUP EXAMINATION, VERSION A PAGE 7. Which of the following is a linear transformation from R to R, that is, T : R R? a) T x x x x 2 = x x 2 x x x 2 x b) T x 2 x = x + x 2 x x 5x 2 x x 2 + x x c) T x x x 2 + 2x x 2 = x x x 2 + 7x x d) T x 2 x = 2x + x x 9 x 2. Given that S = 2 5, 2, v 8 which of the following choices will make S linearly independent? a) v = b) v = c) v = d) v =

8 MATH 22 MAKEUP EXAMINATION, VERSION A PAGE 8. Suppose function T (x) is a linear transformation from R R with the standard matrix A = Which of the following is true? a) T (x) is one-to-one and onto. b) T (x) is onto but not one-to-one. c) T (x) is one-to-one but not onto. d) T (x) is neither one-to-one nor onto.. Which of the matrices below represents the linear transformation T from R 2 to R 2 which reflects points through the line x 2 = x? [ ] a) [ ] b) [ ] c) [ ] d)

9 MATH 22 MAKEUP EXAMINATION, VERSION A PAGE 9 5. Let Find B T A. 2 A = 2, B =. a) b) c) d) [ ] [ ] 6. Let A, B, and C be three invertible n n matrices. Which of the following statements is true? (I) (II) (III) (ABC) = A B C (B T C T ) = (C ) T (B ) T a) (I) and (III) are true, but (II) is false. b) (II) and (III) are true, but (I) is false. c) (I) and (II) are true, but (III) is false. d) All statements (I), (II), and (III) are true. The linear transformation mapping x to Ax is one to one.

10 MATH 22 MAKEUP EXAMINATION, VERSION A PAGE 7. If what is the first row of A? A = 2, 2 6 a) b) 5 c) d) The inverse does not exist. 8. If an n n matrix A is invertible, which of the following is false? a) There is a matrix C such that AC = I. b) A has n pivots when written in echelon form. c) The linear transformation mapping x to Ax is not one-to-one. d) The columns of A are linearly independent.

11 MATH 22 MAKEUP EXAMINATION, VERSION A PAGE 9. Let A be a matrix and let u, v be elements of the nullspace of A. Which of the following are in the nullspace of A? a) u v b) v c) The zero vector d) All of the above 2. Which of the following sets of vectors forms a basis for a subspace of dimension? a),, b),, 2 2 c),, 5 2 5, 2 {[ [ [ d),, ] ] ]}