Alright, so we have a homogeneous system of 8 equations in 10 variables (an 8 x 10 matrix, let's call it A). We have found two solutions that are not multiples of each other (lets call them a and b), and every other solution is a linear combination of them. Can you be certain that any nonhomogeneous equation with the same coefficients has a solution?

I want to say yes, but I'm not sure why. Here's the stuff I know:

  • Our solution for the homogeneous system is span{a, b}.
  • Since there are free variables/the null space is not just 0 we know there are nontrivial solutions.
  • Dim(Null(A))=2
  • $\begingroup$ should be Dim(Null($A$))${}=2$, right? $\endgroup$ – Greg Martin Nov 17 '14 at 4:36
  • $\begingroup$ Do you know a criterion for when any nonhomogeneous equation has a solution? Something related to the rank of $A$, perhaps? $\endgroup$ – Greg Martin Nov 17 '14 at 4:37
  • 1
    $\begingroup$ I know that if the rank=n there is a solution. $\endgroup$ – g.z. Nov 17 '14 at 4:41

You know that it's an 8x10 matrix and that dim Nul A is 2. Here we have n = 10. The rank-nullity theorem says that rank A + dim Nul A = n, so rank A must be 8. In other words, there are 8 pivot columns.

Since your matrix has 8 rows, every row has a pivot, and therefore every solution in $\mathbb{R^8}$ can be represented as a linear combination of the columns.

The general solution of Ax = b can be represented by

(v + linear combinations of solutions to the homogeneous equations)

where v is the particular solution to the nonhomogeneous equation. So yes, a nonhomogeneous equation with the same coefficients has a solution.


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