Check that your exam contains 30 multiple-choice questions, numbered sequentially.

Transcription

1 MATH EXAM SPRING 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 multiple-choice questions, numbered sequentially. Answer Questions 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 EXAM, VERSION A PAGE. Determine h and k such that the solution set of the following system is empty. x + x = k 4x + hx = 8 h =, k b) h =, k = c) h, k d) h, k =. Let v = Span{v,v }?, v = 8, and y = h 5. For what value(s) of h is y in a) h = b) h = 4 c) h = 7 d) All real numbers. Suppose AB = a) A = 5 4 and B = 7 7 b) A = c) A = 4 8 d) A = Find A.

3 MATH EXAM, VERSION A PAGE 4. If det a b c a b c d e f = 8, what is det g + a h + b i + c? g h i d e f a) 8 b) 4 c) 4 d) 8 5. Find the determinant of a) b) c) d) 6. Let A be an invertible n n matrix. Which of the following statements is not necessarily true? a) The equation Ax = b has at least one solution for each b in R n. b) A T is invertible. c) Col A = R n. d) The number is an eigenvalue of A.

4 MATH EXAM, VERSION A PAGE Let A =. Which of the following matrices is similar to A? b) 4 c) d) None of the above. 8. Which of the following is a unit vector in the same direction as v = a) 6 6/ 6? b) 6/6 6/ 6/6 c) /6 / /6 d) v is already a unit vector.

5 MATH EXAM, VERSION A PAGE 5 9. Which of the following statements is always true? a) An n n symmetric matrix has n distinct real eigenvalues. b) An orthogonal matrix is orthogonally diagonalizable. c) If P is an n n matrix with orthogonal columns, then P T = P. d) Every symmetric matrix is orthogonally diagonalizable.. Find the matrix of the quadratic form 8x + 7x x 6x x + 4x x x x. 8 7 b) / c) 4 / 7/ / d) 8 7. Find the equation y = β + β x of the least-squares line that best fits the data points (, ), (, ), (, ), (, ). y =.9 +.4x b) y = x c) y =.4 +.9x d) y = x

6 MATH EXAM, VERSION A PAGE 6. Find the characteristic polynomial of the following matrix: A = 5. 4 a) ( λ)(5 λ)(4 λ) b) (5 λ)(λ λ 4) c) λ + 5λ + 6λ 4 d) 5(λ λ 4). For what value of h are the vectors, 4, h 5 6 linearly independent? a) h b) h c) h = d) h = ( 4. Let T: R R be a linear transformation such that T (). What is T 5? ) = 7 ( and T ) = a) b) c) d) 8 8 4

7 MATH EXAM, VERSION A PAGE 7 5. If A =, find the second column of A. a) b) c) d) 6. Find the coordinate vector of x relative to the basis {v,v } where x =, v = 7 4, v = b) c) d)

8 MATH EXAM, VERSION A PAGE 8 7. What is the rank of a 6 matrix whose null space is 4-dimensional? a) b) c) d) 4 8. Find the eigenvalues of a),, b),, c),, d),, Given the diagonalized matrix A = compute A. + = / / / / / / / /, a) + b) + + c) + d) + + +

9 MATH EXAM, VERSION A PAGE 9. What is the distance between u = and v =? a) b) c) d). Find the orthogonal projection of x onto Span{v,v } where x = 4 4 b) c) d), v =, and v =.

10 MATH EXAM, VERSION A PAGE. If A is an n n square matrix, and I denotes the n n identity matrix, which of the following statements is not necessarily true? If the columns of A form an orthogonal set, then A is an orthogonal matrix. b) If AA T = I, then A is an orthogonal matrix. c) If A is an orthogonal matrix, its determinant must be ±. d) If A is an orthogonal matrix, the linear transformation x Ax preserves lengths and orthogonality.. Which of the following statements is not true? a) An n n matrix that is orthogonally diagonalizable must be symmetric. b) For any matrix A, A T A is symmetric. c) If A is symmetric, any two eigenvectors of A corresponding to different eigenvalues must be orthogonal. d) If A and B are symmetric matrices, AB must also be symmetric. 4. Find the least-squares solution for the system Ax = b where A = / 5/9 / b) /9 / c) 7/9 / d) /9 and b =.

11 MATH EXAM, VERSION A PAGE 5. Let v =, v =, v =, and v = x y. Which of the following z correctly express v as a linear combination of v, v and v? a) v = x + y + z b) v = x + y + z 7 c) v = x + y + z d) v = x y z v + x y v + x + y z 4 v + x y v + x y 7 v + x + y z v 5 v + x y v v + x + y z v 6 v + x y + z v 6 6. Let W = Span{v,v } where v = belong to the orthogonal complement of W? b) c) d) and and and and and v = 6. Which of the following vectors

12 MATH EXAM, VERSION A PAGE 7. Find a basis of the column space ColA for the matrix A = a) The first and second columns b) The first and last columns c) The first, second and third columns d) The fourth and fifth columns 8. Let v = b) c) d) Which of the following vectors is in the same direction as v?

13 MATH EXAM, VERSION A PAGE 9. Let y =, v =. What is the distance from y to the line through and v? a) b) c) d). The vectors 4 and 5 6 form a basis for a subspace W. Use the Gram-Schmidt process 7 to produce an orthogonal basis for W. a) 4, b) 4, c), d) 4, 5 6 7