MATH 2LA3 Midterm Test #1 Midterm Test Duration of test: 50 minutes McMaster University February 9, 2022 Matt Valeriote Last Name: First Name: Student No.: This test has 8 pages and ten questions (some with multiple parts). The last two pages are blank and can be used for your rough work. Please answer all ten questions. Provide justifications for your answers when indicated to do so. For all questions, write your solutions in the space provided for each question. Please be sure to include your student number and initials on each page of this test. A McMaster standard calculator (Casio fx-991 or Casio fx-991 MS) is allowed, but no other aids are. Marks are indicated in the margin; the maximum number of marks is 30. Score Question Score 1 2 3 4 5 6 7 8 9 10 Total MATH 2LA3 Test #1, [5] Student # Initials: Page 2 1. For each of the following statements, determine if they are true or false and circle the correct answer (TRUE or FALSE). Each question is worth 1 mark. Do not provide justifications for your answers. No partial credit will be given for an incorrect answer. There is no penalty for an incorrect answer. (a) If A is an m × n matrix that has rank k, then A has exactly k pivot columns. TRUE FALSE (b) If A is an m × n matrix then the non-zero rows of A form a basis for the row space of A. TRUE FALSE (c) If m ≥ 1 and S = {⃗v1 , ⃗v2 , . . . , ⃗vm } is a set of non-zero vectors from Rn , then some subset of S is a basis for the subspace Span S. TRUE FALSE (d) If ⃗x is an extreme point of the set F of all feasible solutions of a canonical linear programming problem, then ⃗x is an optimal solution of the problem. TRUE FALSE (e) If a canonical linear programming problem does not have an optimal solution, then either the objective function is not bounded on the set of feasible solutions F, or F is empty. TRUE FALSE MATH 2LA3 Test #1, Student # Initials: Page 3 2. Consider the 4 × 5 matrix 1 3 −2 1 3 5 15 1 −28 8 A= 4 12 −1 −17 8 −2 −6 1 7 −6 A can be row-reduced to the matrix 1 0 B= 0 0 [2] 3 −2 1 3 0 1 −3 0 0 0 0 −4 0 0 0 0 (you do not need to show this). In the space below, write down the reduced row echelon form of A (so write down the matrix rref(A)). 3. Using the matrix A from Question #2, determine the rank of A and the nullity of A. Write your answers in the spaces provided below: [2] (a) The rank of A = (b) The nullity of A = . . 4. Using the matrix A from Question #2, [4] (a) find a basis for the column space of A that consists of column vectors of A. List the vectors from this basis in the space below. MATH 2LA3 Test #1, Student # Initials: Page 4 (b) Find a basis for the null space of A and list the vectors of this basis in the space below. 5. Consider the following vectors from R3 : 1 0 6 4 ⃗v1 = −3 , ⃗v2 = 1 , ⃗v3 = −6 , and p⃗ = −1 . 0 2 24 22 [3] Show that the vector p⃗ is in Span{⃗v1 , ⃗v2 , ⃗v3 } by finding scalars a, b, c with p⃗ = a⃗v1 + b⃗v2 + c⃗v3 . Place your answer in the following space: a= ,b= ,c= . You do not need to provide all possible values for a, b, and c. 6. Consider the following 3 × 3 matrix 1 1 1 C = 1 2 2 . 1 2 3 [2] Is the matrix C invertible? Place your answer (YES or NO) in the space below. To receive full credit, provide a justification for your answer. 7. Consider the following canonical linear programming problem: Maximize the objective function f (x, y) = x + 5y subject to the constraints −x + y ≤ 11 x + y ≤ 27 2x + 5y ≤ 90 [3] and x ≥ 0, y ≥ 0. The extreme points of the set of feasible solutions of this problem are: A = (0, 0), B = (0, 11), C = (5, 16), D = (15, 12), and E = (27, 0) (you do not need to show this). Answer the following two questions: MATH 2LA3 Test #1, Student # Initials: Page 5 . To receive full credit, (a) The optimal solution of this problem is (x, y) = provide a brief justification for your solution. (b) The maximum value of the objective function f (x, y) at the optimal solution is 8. Consider the following simplex tableau: x1 x2 x3 x4 x5 x6 M 0 2 4 1 0 2 0 1 2 1 0 0 −3 0 0 1 6 0 1 4 0 0 −2 −1 0 0 5 1 6 8 7 48 [3] (a) Write down the basic feasible solution associated with this simplex tableau. x1 = , x2 = , x3 = , x4 = , x5 = , x6 = . (b) For this basic feasible solution, what is the value of the objective function of the problem? The value is . 9. For this question, use the simplex tableau from the previous question. [3] (a) Identify the entry in the tableau to pivot on in the next step of the simplex method. Provide the row number and column number of this entry: Row number = , Column number = . (b) Write down the simplex tableau that is produced by pivoting on this entry of the given simplex tableau. . MATH 2LA3 Test #1, Student # Initials: Page 6 10. Consider the following canonical linear programming problem: [3] Maximize the objective function f (x1 , x2 ) = 8x1 + 7x2 subject to the constraints x1 + x2 ≤ 17 x1 + 2x2 ≤ 32 5x1 − 12x2 ≤ 10 and x1 ≥ 0, x2 ≥ 0. In the space below, write down the initial simplex tableau for this problem. Note that you do not need to solve this problem, just write down its initial simplex tableau. 1111111111111111111111111111 MATH 2LA3 Test #1, Student # This page intentionally left blank. Initials: Page 7 MATH 2LA3 Test #1, Student # This page intentionally left blank. Initials: Page 8