# Quiz: Linear systems

## This quiz is designed to test your knowledge of systems of linear equations.

Discuss this quiz

(Key: correct, incorrect, partially correct.)

1. What are all the solutions to the system and ?
1. can be arbitrary, and is equal to .
• CORRECT.
2. and are both zero.
3. There are no solutions.
4. and .
• INCORRECT. This is only one of the solutions.
5. and .
• PARTIALLY. These are valid equations, but this does not quite answer the question.
6. and , or and .
7. and .
2. What are all the solutions to the system and ?
1. and .
• CORRECT.
2. and .
3. There are no solutions.
4. There are infinitely many solutions.
5. and , or and .
6. can be arbitrary, and is equal to .
7. can be arbitrary, and is equal to .
3. What are all the solutions to the system and ?
1. There are no solutions.
• CORRECT.
2. There are infinitely many solutions.
3. and .
4. can be arbitrary, and is equal to .
5. can be arbitrary, and is equal to .
6. This question cannot be answered correctly.
7. .
• INCORRECT. This does not answer the question.
4. If one is solving three linear equations involving two unknowns, what happens?
1. Usually there will be no solution, but occasionally there will be one or more solutions.
• CORRECT.
2. There will never be a solution.
3. There will always be a solution.
4. There will always be infinitely many solutions.
5. Usually there will be one solution, but occasionally there will be no solutions or infinitely many solutions.
6. Usually there will be infinitely many solutions, but occasionally there will be one or no solutions.
7. Anything can happen.
8. There will always be exactly one solution.
5. If one is solving two linear equations involving three unknowns, what happens?
1. Usually there will be no solution, but occasionally there will be one or more solutions.
2. There will never be a solution.
3. There will always be a solution.
4. There will always be infinitely many solutions.
5. Usually there will be one solution, but occasionally there will be no solutions or infinitely many solutions.
6. Usually there will be infinitely many solutions, but occasionally there will be one or no solutions.
7. Usually there will be infinitely many solutions, but occasionally there will be no solutions.
• CORRECT.
8. There will always be exactly one solution.
6. If one is solving three linear equations involving three unknowns, what happens?
1. Usually there will be no solution, but occasionally there will be one or more solutions.
2. There will never be a solution.
3. There will always be a solution.
4. There will always be infinitely many solutions.
5. Usually there will be one solution, but occasionally there will be no solutions or infinitely many solutions.
• CORRECT.
6. Usually there will be infinitely many solutions, but occasionally there will be one or no solutions.
7. Anything can happen.
8. There will always be exactly one solution.
7. What is the complete relationship between homogeneous linear systems of equations, and the zero solution (all unknowns equal to zero)?
1. The zero solution is always a solution to homogeneous linear systems, and never a solution to inhomogeneous linear systems.
• CORRECT.
2. The zero solution is always a solution to both homogeneous and inhomogeneous linear systems.
3. The zero solution is always a solution to homogeneous linear systems, but could also be a solution to inhomogeneous linear systems.
4. The zero solution is never a solution to inhomogeneous linear systems, and may or may not be a solution to homogeneous linear systems.
5. The zero solution can be a solution to both homogeneous and inhomogeneous linear systems, but only if the equations are solvable.
6. If a solution to a homogeneous linear system exists at all, it will be the zero solution.
7. If a solution to a homogeneous linear system exists at all, then the zero solution will be a solution.
• PARTIALLY. Homogeneous linear systems always have at least one solution, namely the zero solution.
8. If one is solving three homogeneous equations involving two unknowns, what happens?
1. Usually the zero solution is the only solution, but occasionally one has more solutions.
• CORRECT.
2. Usually one has no solutions, but occasionally one has one or infinitely many solutions.
3. The zero solution is the only solution.
4. There are no solutions.
5. One can get different answers, depending on how you approach the problem.
6. One has infinitely many solutions, including the zero solution.
7. One usually has infinitely many solutions, but occasionally one just has only the zero solution.
9. If one is solving two homogeneous equations involving three unknowns, what happens?
1. Usually the zero solution is the only solution, but occasionally one has more solutions.
2. Usually one has no solutions, but occasionally one has one or infinitely many solutions.
3. The zero solution is the only solution.
4. There are no solutions.
5. One can get different answers, depending on how you approach the problem.
6. One has infinitely many solutions, including the zero solution.
• CORRECT.
7. One usually has infinitely many solutions, but occasionally one just has only the zero solution.
10. If one is solving three homogeneous equations involving three unknowns, what happens?
1. Usually the zero solution is the only solution, but occasionally one has more solutions.
• CORRECT.
2. Usually one has no solutions, but occasionally one has one or infinitely many solutions.
3. The zero solution is the only solution.
4. There are no solutions.
5. One can get different answers, depending on how you approach the problem.
6. One has infinitely many solutions, including the zero solution.
7. One usually has infinitely many solutions, but occasionally one just has only the zero solution.
11. If a linear system has three equations in four unknowns, then
1. The rank of this system can be any number from zero to three.
• CORRECT.
2. The rank of this system is three.
3. The rank of this system is four.
4. The rank of this system can be any number from zero to four.
5. The rank of this system can be three or four.
6. The rank of this system is twelve.
7. The rank of the system is at least three.
12. If a linear system has four equations in three unknowns, then
1. The rank of this system can be any number from zero to three.
• CORRECT.
2. The rank of this system is three.
3. The rank of this system is four.
4. The rank of this system can be any number from zero to four.
5. The rank of this system can be three or four.
6. The rank of this system is twelve.
7. The rank of the system is at least three.
13. If a linear system has four unknowns and has rank three, then
1. There are infinitely many solutions, unless the system is inconsistent, in which case there are no solutions.
• CORRECT.
2. There are infinitely many solutions.
3. There are no solutions (system is inconsistent).
4. There is exactly one solution.
5. There is either one solution or infinitely many solutions.
6. There is either one solution or no solution.
7. Anything can happen.
14. If a linear system has four unknowns and has rank four, then
1. There are infinitely many solutions, unless the system is inconsistent, in which case there are no solutions.
2. There are infinitely many solutions.
3. There are no solutions (system is inconsistent).
4. There is either one solution or no solution.
• CORRECT.
5. There is either one solution or infinitely many solutions.
6. There is exactly one solution.
7. Anything can happen.
15. If a linear system has four unknowns, four equations and has rank four, then
1. There are infinitely many solutions, unless the system is inconsistent, in which case there are no solutions.
2. There are infinitely many solutions.
3. There are no solutions (system is inconsistent).
4. There is exactly one solution.
• CORRECT.
5. There is either one solution or infinitely many solutions.
6. There is either one solution or no solution.
7. Anything can happen.