# Given a homogeneous system, what can we say about a similar but nonhomogeneous system?

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
• should be Dim(Null($A$))${}=2$, right? – Greg Martin Nov 17 '14 at 4:36
• Do you know a criterion for when any nonhomogeneous equation has a solution? Something related to the rank of $A$, perhaps? – Greg Martin Nov 17 '14 at 4:37
• I know that if the rank=n there is a solution. – g.z. Nov 17 '14 at 4:41

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.