Question 1. The homogeneous equation Ax = 0 has the trivial solution if and only if the equation has at least one free variable.
Answer: False. Every homogeneous equation has the trivial solution x = 0.
Question 2. The parametric equation x = p + tv described a line through v parallel to p.
Answer: False. This is a line through p parallel to v. Try t = 0.
Question 3. The solution set of Ax = b is the set of all vectors of
the form w = p + v, where v is any solution of Ax = 0, and Ap = b.
Answer: True. See Theorem 6, page 52.
Question 4. If x is a nontrivial solution of Ax = 0, then every entry in x is nonzero.
Answer: False. If x is not equal to the zero vector, and Ax = 0, then x is a nontrivial solution. The trivial solution has all entries 0. If some entries are zero but not all, then x is not = 0.
Question 5. The equation Ax = b is homogeneous if the zero vector is a solution.
Answer: True. If 0 is a solution, then A0 = b. But A0 = 0 for any matrix
A, so b = 0.
| Question 6. A is a 3x3 matrix with 3 pivot positions. Select all the statements which must be true for this A. | Ax = 0 has a nontrivial solution. | False |
| Ax = b has at least one solution for every possible b. | True | |
| Question 7. A is a 3x3 matrix with 2 pivot positions. Select all the statements which must be true for this A. | Ax = 0 has a nontrivial solution. | True |
| Ax = b has at least one solution for every possible b. | False | |
| Question 8. A is a 3x2 matrix with 2 pivot positions. Select all the statements which must be true for this A. | Ax = 0 has a nontrivial solution. | False |
| Ax = b has at least one solution for every possible b. | False | |
| Question 9. A is a 2x4 matrix with 2 pivot positions. Select all the statements which must be true for this A. | Ax = 0 has a nontrivial solution. | True |
| Ax = b has at least one solution for every possible b. | True |