Science:Math Exam Resources/Courses/MATH152/April 2012/Question 06 (d)

MATH152 April 2012

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Question 06 (d)

This question requires little or no calculations. Consider a linear system of 3 equations with 5 unknowns. Answer True or False to each of the following statements (no justification required). (1) It always has at least one solution. (2) There is either no solution or infinitely many solutions. (3) If a solution exists, then there is precisely a 2-parameter family of solutions. (4) If the coefficient matrix has rank k, then the associated homogeneous system as a k-parameter family of solutions.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Try to determine a counter-example first. If you find it difficult to determine one, the answer may be true.

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. (1) It always has at least one solution. (FALSE) (2) There is either no solution or infinitely many solutions. (TRUE) (3) If a solution exists, then there is precisely a 2-parameter family of solutions. (FALSE) (4) If the coefficient matrix has rank k, then the associated homogeneous system as a k-parameter family of solutions. (FALSE) The justification for each answer is given below. (1) is FALSE. Consider the following counter-example of three equations with five unknowns: $x=1,\quad x=2,\quad x+y+z+u+v=3$ Clearly, this system is inconsistent. Hence (1) is FALSE by counter-example. (2) is TRUE. As we saw in (1), it is possible that there is no solution. For the second part of the statement, assume that a solution exists. This solution cannot be unique since any row-reduced echelon form of the coefficient matrix admitting a solution has at least two free parameters (i.e. Rank of the matrix is at most 3, but there are 5 parameters). Hence, if any solution exists, then infinitely many solutions must exist. Therefore (2) is true. (3) is FALSE. Consider the following example: ${\begin{aligned}x+y+z+u+v&=0\\x+y+z+u+v&=0\\x+y+z+u+v&=0\end{aligned}}$ In this example, there is a four-parameter family of solutions since the associated coefficient matrix has rank 1. (4) is FALSE. See (1) and (3) for counterexamples.

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