Suppose a nonhomogeneous system of 15 linear equations in 17 ...

[DelorasTo]

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.

[nanny]

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.

[DelorasTo]

Thank you!

[nanny]

Happy to help