# Possibilities of the dimension of the solution space

If we have a $n$ linear equations and $m$ unknowns, what are the possibilities for the dimension of the solution space when $n > m$ and when $n < m$ for a homogeneous system of equations? How do I approach this problem? I know it has something to do with the rank-nullity theorem.

• what is the dimension of the solution space in terms of the rank and nullity? Commented Mar 15, 2017 at 17:44
• @Callus if you consider the the system of equations as a linear transformation, then the kernel would represent all solutions to the homogenous system of equations, I'm unsure what to do next though
– AAZ
Commented Mar 15, 2017 at 19:03
• right, so the dimension of that space is the nullity. What are the possible ranks if $n>m$ and $n<m$? Commented Mar 15, 2017 at 19:18
• @Callus this is probably wrong, but wouldn't the rank be min(m,n)?
– AAZ
Commented Mar 15, 2017 at 19:19
• That's the highest the rank can be. Commented Mar 15, 2017 at 19:21