Science:Math Exam Resources/Courses/MATH152/April 2022/Question A12

MATH152 April 2022

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Question A12

Consider the system of linear equations ${\textstyle Ax=b}$, where ${\textstyle A}$ is a ${\textstyle 5\times 7}$ matrix of real numbers, ${\textstyle b}$ is a ${\textstyle 5\times 1}$ vector of real numbers, and ${\textstyle x}$ is a ${\textstyle 7\times 1}$ vector of unknowns. Which of the following could describe the collection of all solutions ${\textstyle x}$? List all that are possible for some ${\textstyle A}$ and ${\textstyle b}$. no solutions precisely one solution precisely 5 solutions precisely 7 solutions infinitely many solutions, forming a line infinitely many solutions, forming a plane

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Go through the options one-by-one and consider the extreme situations. For example, what if every entry in the matrix is 0? What if the vector $b$ is 0? Does this suggest what you should focus on to understand the problem?

Hint 2
Consider the associated homogeneous equation $A$: $$Ax=0.$$ What can you say about the solutions here? Unless $b$ is the $0$ vector, solutions of the homogenous equation are not solutions of the original equation. However, given a solution of the original equation, can you use knowledge about the space of homogenous solutions to say something about the full set of solutions?

Checking a solution serves two purposes: helping you if, after having used all the hints, 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

Following the first hint, we see that option 1 is possible: consider the possibility that $A$ is the 0 matrix and $b$ is not zero. If, instead, $b$ is the zero vector, then every $x\in \mathbb {R} ^{7}$ solves the equation. Now let us consider if option 2 is possible. We need to be in a situation where we have at least one vector $x_{1}$ that satisfies $Ax_{1}=b.$ Following the second hint, consider the homogenous equation $$Ax=0,$$ which corresponds to a system of equations $${\begin{cases}a_{1,1}x_{1}+a_{1,2}x_{2}+a_{1,3}x_{3}+\dots +a_{1,7}x_{7}&=0\\a_{2,1}x_{1}+a_{2,2}x_{2}+a_{2,3}x_{3}+\dots +a_{2,7}x_{7}&=0\\\quad \vdots &\\a_{5,1}x_{1}+a_{5,2}x_{2}+a_{5,3}x_{3}+\dots +a_{5,7}x_{7}&=0\end{cases}}$$ This system of equations has more variables than equations. Since there are two fewer equations than variables, there will be (at least) two free variables when solving the system, so the vectors $x_{0}$ that solve the homogeneous equation form a subspace of dimension at least 2. Now, given a solution $x_{1}$ of the original equation and a solution $x_{0}$ of the homogenous equation, $x_{1}+x_{0}$ is another solution of the original equation. Since we have a subspace of homogenous solutions of dimension at least 2, it follows that we also have at least a 2-dimensional set of solutions to the original equation. This shows that options 2, 3, 4, 5 are not possible, so only options 1 and 7 can describe the solutions to $Ax=b$.