Science:Math Exam Resources/Courses/MATH152/April 2016/Question A 17
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Question A 17 

Consider a linear system with 7 equations for 8 unknowns. Circle all possible types of solution sets that could result:

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 matrix representation of the system. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We can write a linear system with 7 equations for 8 unknowns as a matrix M with 7 rows and 8 columns. If the entries of M are all zeros. b is a vector with entries all ones. Mx = b has no solution, so (a) is possible. The system has unique solution if the kernel is zero. Because the matrix M has 8 columns and 7 rows, its column spaces are linear dependent. There is a nonzero vector in the null space, so (b) is wrong. By the ranknullity theorem, Nullity + rank(M) = 8 , the dimension of kernal is in the range [1,8]. When the matrix has rank 0 , the dimension of kernal is 8 , which means the system has 8 distinct solutions. (c) is possible If dimension of kernal is 2, that's the case of solutions with 2 parameters , and likewise the solutions of 1 parameter. Thus, (d)(e) is possible Thus, the possible results are . 
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