Fall 2016 MATH*1160 Final Exam

Transcription

1 Fall 2016 MATH*1160 Final Exam Last name: (PRINT) First name: Student #: Instructor: M. R. Garvie Dec 16, 2016 INSTRUCTIONS: 1. The exam is 2 hours long. Do NOT start until instructed. You may use blank areas of this exam booklet for rough work. Notes or books are not permitted during the exam. 2. You may use a calculator, but not a graphing calculator that supports matrix algebra. No other electronic equipment is permitted. Turn off or mute your cell phone! The use of cell phones during the exam is prohibited. 3. The test consists of 40 equally-weighted (independent) multiple choice questions. Answer the multiple choice questions on the computer score sheet (circle your answers on the exam paper). In each question choose the answer that best fits the question. Make sure you have a complete exam booklet. 4. Fill in the computer score sheet in pencil; make sure you include your name and student ID number, and your UoG address, but you don t need to enter the section number. Also fill in your name and student number at the top of this exam booklet. 5. If you copy the work of your neighbour this is considered Academic Misconduct and will be reported to the appropriate university authority. 6. Hand in the entire exam booklet and your computer score sheet.

2 1. Choose the statement that best describes the solvability of a homogeneous system of linear equations. (A) Is inconsistent (B) Has only the trivial ( zero ) solution (C) Has either just the trivial ( zero ) solution, or an infinite number of solutions (D) Has either a unique solution, no solution, or an infinite number of solutions (E) Has an infinite number of solutions 2. The linear system of equations has (A) No solution (B) A unique solution (C) An infinite number of solutions (D) Exactly two solutions (E) A trivial solution { 3x 2y = 4 6x 4y = 8 2

3 Let A = and B = 0 1 2, then (A T B) T is given by (A) (B) (C) (D) (E) The numbers x, y, z, w R satisfy the following matrix equation: ( ) ( ) x y 2x 1 =. z z w 4 2 Then (x, y, z, w) = (A) ( 1, 1/2, 4, 6) (B) (1, 1/2, 4, 2) (C) (1, 1/2, 4, 6) (D) ( 1, 1/2, 4, 2) (E) No solution 3

4 5. Let ( ) 2 4 A = 1 1 and x = ( ) 4. 1 Consider the matrix equation Ax = kx where k R. Then the scalar k (A) Does not exist (B) = 2 (C) = 1 (D) = 3 (E) = The value(s) of k R such that (ka) T (ka) + 1 = 0, where A = 2 is 0 (A) ±1/ 8 (B) +1/8 (C) ±1/ 3 (D) Not a real number (E) 1/8 4

5 7. Let A, B and C be matrices and r R. Then ((A T + rb)c) T = (A) C T (A T + rb T ) (B) (A T + rb)c T (C) C T (A + rb T ) (D) C T (A + r T B T ) (E) (A + rb T )C T 8. Consider the matrix equation ACx = b where ( ) ( ) A 1 =, C 1 = The solution x is given by ( ) 19 (A) 1 ( ) 7 (B) 1 ( ) 19 (C) 4 ( ) 7 (D) 4 ( ) 19 (E) 3, b = ( )

6 9. Calculate (ca) 1 when c = 2 and A 1 = ( ) 1/2 0 (A) 1 3/2 ( ) 2 0 (B) 4 6 ( ) 2 0 (C) 4/3 2/3 ( ) 1/2 0 (D) 1/3 1/6 ( ) 1/2 0 (E) 4 3/2 ( ) The matrix is (A) Not in reduced row echelon form (B) In reduced row echelon form (C) In row echelon form, but not reduced echelon form (D) In reduced row echelon form, but not in row echelon form (E) Not in row echelon form 6

7 11. The reduced row echelon form of ( ) (A) ( ) (B) ( ) (C) ( ) (D) ( ) (E) ( ) is The linear algebraic system of equations associated with the following augmented matrix is (A) (B) (C) (D) { x1 3x 2 + 2x 3 = 1 5x 2 x 3 = 2 2x 1 x 2 + x 3 = 0 3x 1 + 5x 2 2x 3 = 0 x 1 = x 1 x 2 x 3 2x 1 x 2 = 1 3x 1 + 5x 2 = 2 x 1 = 0 (E) None of the above 1 = 2 0 7

8 13. The solution of the system is (A) = {(α, α 1, 3α) α R} (B) = (x 1, x 2, x 3 ) = (3, 0, 1) (C) Does not exist { x1 + 2x 2 + x 3 = 2 (D) = {(3α, α 1, 3α + 4) α R} (E) = {(3α, α 1, α) α R} x 2 x 3 = Let A be an n-by-n matrix. Which statements below is equivalent to the statement: The reduced row echelon form of A is I n (I n is the n n identity matrix.) (A) A is singular (B) The homogeneous system Ax = 0 has a nontrivial solution (C) The non-homogeneous system Ax = b has a unique solution for every right hand side vector b (D) A = 0 (E) The row echelon form of A is equal to the reduced row echelon form of A 8

9 15. If A = a c (A) 5/3 (B) 15 (C) 5 (D) 1/15 (E) 15 b d = 5, then B = 3b + a 3d + c a c is equal to 16. Let A be a 4 4 matrix such that A = 4. Then 2A 1 is equal to (A) 4 (B) 4 (C) 1/2 (D) 1/2 (E) 8 9

10 17. Let A = (a ij ) be an n n upper triangular matrix with n a ii = 2 (product of the diagonal entries). Then the homogeneous linear system of equations in matrix form Ax = 0 has (A) Exactly n solutions (B) No solution (C) An infinite number of solutions (D) An infinite number of solutions, or no solution (E) Only the trivial solution If A = then the cofactor expansion along the 2nd column yields A = (A) (B) (C) (D) (E) i=1 10

11 19. Let A = ( ) 1 2. Then the inverse of A is given by 3 4 ( ) 2 1 (A) 3/2 1/2 ( ) 4 2 (B) 3 1 ( ) 1 1/2 (C) 1/3 1/4 ( ) 2 1 (D) 3/2 1/2 ( ) 1/2 1 (E) 3/ Let u = x + y = (A) 5/2 (B) 2 (C) 1/2 (D) 3/2 (E) 1 ( ) 2, v = 4 ( ) 3 and w = 3 ( ) 4. If xu + yv = w, (x, y R), then 5 11

12 21. The vectors u, v and w are illustrated below. OABC is a parallelogram. From the diagram we see that u + v + w = (A) u (B) 0 (the zero vector) (C) u v (D) u + v (E) 2u 12

13 22. The vectors OA = u, OE = v and OD = w are represented below. OECD and OABC are parallelograms. The vector AC written in terms of the given vectors is (A) u + w (B) v + w u (C) u + v + w (D) u v + w (E) w u 13

14 23. Let V be a vector space with the operations of scalar multiplication and vector addition. Which rule below is NOT an axiom of a vector space, where u, v, w V and c, d R: (A) (c d) u = c (d u) (B) (c + d) u = c u d u (C) c (u v) = c u c v (D) u v = v u (E) (u v) w = u (v w) 24. Let W be a subspace of a vector space V. Here and represent vector addition and scalar multiplication respectively. Then it follows that (A) W is a vector space with respect to the same operations of vector addition and scalar multiplication as for V (B) the zero vector 0 V is also the zero vector of W (C) W is a subset of V (D) u, v W (E) all of the above = u v W 14

15 25. Let A be an n n matrix and b be an n 1 vector. Consider the following set: Which statement below is CORRECT? (A) W is not a subspace of R n W = {x R n Ax = b, b 0}. (B) W is closed with respect to scalar multiplication (C) W is a subspace of R n (D) W is closed with respect to vector addition (E) W is the null-space of A 26. Suppose the vectors v 1, v 2,..., v k belong to a vector space V. Which statement below is NOT necessarily true. (A) The span of the vectors v 1, v 2,..., v k is given by the set of all linear combinations of these vectors. (B) span{v 1,..., v k } = {a 1 v 1 + a 2 v a k v k a i R} (C) The span of the vectors v 1, v 2,..., v k equals V (D) The span of the vectors v 1, v 2,..., v k is a subspace of V (E) If u, v span{v 1,..., v k } then u + v span{v 1,..., v k } 15

16 27. In the vector space P 2 (set of all polynomials of degree 2) the argument needed to determine if the vector v = 3t 2 + t belongs to the span of {2t 2 t + 1, t 2 1, t + 1} yields the following augmented matrix: (A) (B) (C) (D) (E) Suppose the vectors v 1, v 2,..., v k in a vector space V are linearly dependent. Identify the INCORRECT statement. (A) There exists scalars α i, not all zero, such that k α i v i = 0. i=1 (B) There does not exist a linear relationship between the vectors v 1, v 2,..., v k (C) At least one vector in the span of the vectors v 1, v 2,..., v k is effectively redundant (D) Some of the vectors v 1, v 2,..., v k can be written as a linear combination of the other vectors (E) The vectors are not linearly independent 16

17 29. Which statement concerning the vectors {( ) 1, 2 ( )} 2 is INCORRECT? 3 (A) The given vectors are linearly independent because ( ) 1 2 (B) The given vectors are linearly independent because A := is row 2 3 equivalent to I 2 (the 2 2 identity matrix) ( ) 1 2 (C) With A := the homogeneous system of linear equations Ax = 0 has 2 3 only the trivial solution and thus the given vectors are linearly independent (D) The given vectors are linearly independent because they lie on the same line passing through the origin (E) None of the above 30. The dimension of the vector space P n (set of all polynomials of degree n) is (A) n (B) n 1 (C) n + 1 (D) n 2 (E) n

18 31. Which statement concerning the set S = 0, 3, is CORRECT? (A) span S is a subset of (but NOT equal to) R 3 (B) S is a basis for R 3 (C) S spans R 3, but S is not linear independent (D) S is linear independent, but does not span R 3 (E) dim span S < 3 and thus S cannot be a basis for R The equation of the ( straight ) line passing through the origin that corresponds to 1 2 the null-space of is: 3 6 (A) y = 1 2 x (B) y = 2x (C) y = 2x + 1 (D) y = 2x + 1 (E) None of the above 18

19 The dimension of the null-space of A = is (A) 2 (B) 3 (C) 4 (D) 1 (E) The rank of the matrix A = is (A) 3 (B) 2 (C) 1 (D) 4 (E) 0 19

20 35. Let A be an n n matrix, and b and x be vectors of size n 1. Which statement below is NOT equivalent to the statement rank A = r < n? (A) The system Ax = b has either no solution or an infinite number of solutions (B) A = 0 (C) A is NOT row reducable to I n (D) The system Ax = 0 has a non-trivial solution (E) A is nonsingular 36. Let u = ( 1, 0, 2, x) and v = (1, 1, 2, 4) where x R. The value of x such that u and v are orthogonal is (A) 0 (B) 5/4 (C) 1 (D) 4/5 (E) 2/3 20

21 37. Consider an inner product (, ) defined on a (real) vector space V. Let u, v, w V and c R. Which axiom below is NOT part of the definition of an inner product? (A) (u, v) = (v, u) (B) (u, v) 0 with equality iff u = v (C) (u + v, w) = (u, w) + (v, w) (D) (cu, v) = c(u, v) (E) None of the above is incorrect 38. Consider the standard inner product (u, v) := u v, where u, v R 3. If 1 u = 2 and 1 v = 0, 3 2 then the Cauchy-Schwarz Inequality yields an upper bound on u v to be (A) 70 (B) 5 (C) 18 (D) 70 (E) 18 21

22 39. The eigenvalues of the matrix (A) 1 and 4 (B) 1 and 5 (C) 0 and 5 (D) ±5 (E) 0 and 5 ( ) 1 2 are Suppose a matrix B( ) has eigenvalues ( λ 1 ) = 2 and λ 2 = 2 with associated 1 1 eigenvectors x 1 = and x 2 2 = respectively. The matrices P and D 2 such that D = P 1 BP are given by ( ) ( ) (A) D = and P = ( ) ( ) (B) D = and P = ( ) ( ) (C) D = and P = ( ) ( ) (D) D = and P = (E) None of the above END OF TEST 22