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

Consider the system of linear equations , where is a matrix of real numbers, is a vector of real numbers, and is a vector of unknowns. Which of the following could describe the collection of all solutions ? List all that are possible for some and .

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 

Consider the dimensions of the system and recall that an matrix has rows and columns. Is the system a short/wide system, or a tall/narrow system? In other words, are there more columns than rows, or more rows than columns? 
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.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. This system has more variables than equations, so it will either have no solutions or infinitely many solutions. To see why the system could have 0 solutions, consider the possibility that is a matrix of 0s and is a nonzero vector. Suppose now that there is at least one solution. Since there are two fewer equations than variables, there will be two free variables when solving the system, so the solutions will form a plane. (If there were only one free variable, they would form a line.) 