Dimension of the Solution Set of a System of Linear Equations

Question

I've been stuck on this question for a while now

Let E be a system of m linear equations and n unknowns. Assume E has at least one solution. Then dim(S(E)) is :

a) $\leqslant$ n-m

b) = n-m

c) $\geqslant$ n-m

d) $\leqslant$ m-n

e) = m-n

f) $\geqslant$ m-n

Could you also help me understand the reasoning behind the answer? Thank you in advance

Answer 1

Hint:

Let $r$ be the rank of the system, i.e. the rank of its matrix. We know two things:

1. $ \;r\le\min(m,n)$,
2. $r$ is the codimension of the solution set.

Can you conclude?