Solutions to a system of Linear Equations

[Treadstone 71]

"Show that the number of distinct solutions of a system of linear equations (any number of equations and unknowns) over a field Z_p is either 0, or a power of p."

I don't know where to start. Suppose there are n unknowns, if only I can show that the solution space is a subspace of Z_p ^n, then it's easy. But I can't seem to do it. Any hints?

 

[mathmike]

what are your thoughts on the question? Have you any ideas on where to start?

 

[JasonRox]

Do you know anything about groups?

I think that could help at looking at it.

Another question to ask yourself if you don't know the above is...

...what characteristics must p have for Z_p to be a field?

 

[shmoe]

I don't know where to start. Suppose there are n unknowns, if only I can show that the solution space is a subspace of Z_p ^n, then it's easy. But I can't seem to do it. Any hints?

It's not always a subspace, but you should know something about the structure of the solutions. (if the Z_p is causing problems, what does it look like in the real case?)

 

[Treadstone 71]

I've shown that the set of solutions to the HOMOGENEOUS equivalent of the system of equations is a subspace of Z_p ^n. I've also shown that if the system of equations is not homogeneous, and suppose that there is one solution v, then any other solution is an element of the direct sum of v + null T, which cardinality is a power of p because it's a subspace.