how would you do this proof?

[eldercato]

I thought I was on the right track, several times, but can't seem to do this... I'm wondering if i've taken enough lin alg to answer this, any ideas would be extremely helpful!

Let A be an n x m matrix. Prove that the equation Ax = b has a solution if and only if
<b,v> = 0 for all v in the nullspace of A.

[HallsofIvy]

I thought I was on the right track, several times, but can't seem to do this... I'm wondering if i've taken enough lin alg to answer this, any ideas would be extremely helpful!

Let A be an n x m matrix. Prove that the equation Ax = b has a solution if and only if
<b,v> = 0 for all v in the nullspace of A.
One difficulty with your not showing what work you HAVE done is that we don't know what concepts you have to work with. In fact, I can see one immediate problem with what you have written. If A is an m by n matrix, then it represents a linear transformation from R^n to R^m. The "null space" of A, the set of all vectors, v, such that Av= 0 is a subspace of R^n. In order that Ax= b even make sense b must be in R^m. It's not clear what you mean by "<b,v>" when b and v are in different spaces.

I'm going to need to think about this and I don't have the time right now. I'll try to get back to it.