Solutions to Linear Equations

Date: 09/21/2001 at 05:27:22
From: Nanda
Subject: Linear equations

Dear Dr. Math,

Can you specify the conditions under which a a system of linear 
equations has infinitely many solutions, a unique solution and no 
solution?

Date: 09/21/2001 at 11:51:37
From: Doctor Rob
Subject: Re: Linear equations

Thanks for writing to Ask Dr. Math, Nanda.

Let S be the field from which the coefficients and solution values
are drawn. Let V be the number of variables, and E the number of
*independent* equations. Then:

   1. If E > V, the number of solutions is 0.
   2. If E = V, the number of solutions is 1.
   3. If E < V and S is finite, then the number of solutions is
      a power of the size of S.
   4. If E < V and S is infinite, then the number of solutions is
      infinite.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/