# Determine the number of solutions for a homogeneous system

1. Dec 4, 2011

### anu914

Hi all,

I would like to know how to determine the number of solutions for a Homogeneous system.
Ax = 0

A is a m*n matrix and x is a n*1 vector.

There are m equations and n unknowns. I'd like to know how to determine the number of solutions to this system.

2. Dec 5, 2011

### HallsofIvy

Staff Emeritus
First, thinking of the m rows as vectors, determine how many of them are independent (the "rank" of A). Call that number m'. By the "rank-nullity" theorem, the nullity, the dimension of kernal(A), will be n- m'. The number of (independent) solutions will be 1 if $n- m'\le 0$, and n- m' if it is greater than 0.

3. Dec 5, 2011

### anu914

Thanks a lot for the answer...