Midterm1-PracticeTest-Solutions

Midterm1-PracticeTest-Solutions - Math 331 Midterm 1...

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Unformatted text preview: Math 331 Midterm 1 practice problems This is a list of practice problems for the first midterm in Math 331 (Wylie). The actual exam will be 10 problems long with each problem worth 10 points. The first problem on the exam will be similar to the first practice problem and will be a true/false with 5 parts. You will not need to show any work on this problem. 1. True of False: (a) If A is a 5 7 matrix with 5 pivot columns, then there is only one solution to A x = 0. False. (b) If 2 v 1 + v 2 + 6 v 3 = 0 and A is the matrix with columns v 1 , v 2 , and v 3 then there are infinitely many solutions to the equation A x = 0. True. (c) If A x = 0 has more than one solution then A x = b has more than one solution for every b . False. (d) There is a 2 3 matrix such that A x = 0 has only the trivial solution. False. (e) If the the columns of a 4 4 matrix span R 4 then the linear system A x = b is consistent for every b . True. 2. Find the solution to the system of linear equations using the row reduction method. x + y + z = 6 x- y = 0 x- z = 0 Solution: 1 1 1 6 1- 1 1- 1 0 II- I--- III- I 1 1 1 6- 2- 1- 6- 1- 2- 6 II (- 2)----- 1 1 1 6 1 1 / 2 3- 1- 2- 6 I- II---- III + II 1 0 1 / 2 3 0 1 1 / 2 3 0 0- 3 / 2- 3 III (- 2 / 3)------ 1 0 1 / 2 3 0 1 1 / 2 3 0 0 1 2 I- (1 / 2) III------- II- (1 / 2) III 1 0 0 2 0 1 0 2 0 0 1 2 So the solution is x = y = 2. 3. Determine h and k such that the solution to set of the system equations x 1 + 4 x 2 = h 2 x 1 + kx 2 = 4 (a) is empty. (b) contains a unique solution. (c) contains infinitely many solutions. Solution: 1 4 h 2 k 4 II- 2 I--- 1 4 h k- 8 4- 2 h (a) The solution set is empty if k = 8 and h 6 = 2. (b) The solution set contains a unique solution if k 6 = 8. (c) The solution set is infinite if k = 8 and h = 2. 4. For what values of k will the system of equations- x 1- x 2 + x 3 = 2 kx 1- 9 x 2 = 5 3 x 1 + 2 x 2 + x 3 = 0 be inconsistent? Solution: - 1- 1 1 2 k- 9 0 5 3 2 1 0 II + kI---- III +3 I - 1- 1 1 2- 9- k k 5 + 2 k- 1 4 6 II III---- - 1- 1 1 2- 1 4 6- 9- k k 5 + 2 k III- (9+ k ) II------- - 1- 1 1 2- 1 4 6- 36- 3 k- 49- 4 k The system will be inconsistent when k =- 12. 5. Write the vector b = 1 as a linear combination of the vectors v 1 = 1 1- 1 , v 2 = 1- 1 1 , v 3 = 1- 1- 1 Solution: We need to solve the linear system 1 1 1 1- 1- 1 1- 1 1- 1 0 III + I--- II- I 1 1 1- 2- 2 1 2 II (- 2)----- 1 1 1 0 1 1- 1 / 2 0 2 0 I-...
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Midterm1-PracticeTest-Solutions - Math 331 Midterm 1...

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