0
$\begingroup$

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
$\endgroup$
  • $\begingroup$ 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
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.