Homework Clinic
Mathematics Clinic => Calculus => Topic started by: DelorasTo on Jun 18, 2018
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Suppose a nonhomogeneous system of 15 linear equations in 17 unknowns has a solution for all possible constants on the right side of the equation.
Is it possible to find 3 nonzero solutions of the associated homogeneous system that are linearly independent? Explain.
Question 2
A mathematician has found 5 solutions to a homogeneous system of 40 equations in 42 variables. The 5 solutions are linearly independent and all other solutions can be constructed by adding together appropriate multiples of these 5 solutions.
Will the system necessarily have a solution for every possible choice of constants on the right side of the equation? Explain.
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Answer to Question 1
No.
Since every nonhomogeneous equation Ax = b has a solution, Col A spans . So Dim Col A = 15. By the Rank Theorem, dim Nul A = 17 - 15 = 2. So the associated homogeneous system does not have more than 2 linearly independent solutions.
Answer to Question 2
No.
Let A be the 40 42 coefficient matrix of the system. The 5 solutions are linearly independent and span Nul A, so dim Nul A = 5. By the Rank Theorem, dim Col A = 42 - 5 = 37. Since 37 < 40, Col A does not span . So not every nonhomogeneous equation Ax = b has a solution.
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Thank you!
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Happy to help