is a $\displaystyle m\times n$ matrix and $\displaystyle T:\mathbb{R}^{n}\rightarrow \mathbb{R}^{m}$ is a linear transformation $\displaystyle T(x)=Ax$

1)Show that $\displaystyle Ax=b$ with $\displaystyle m<n$ either has no solution or has infinitely many

2)Show that $\displaystyle Ax=b$ with has rank$\displaystyle m$ always has a solution

3)Show that $\displaystyle Ax=b$ with has rank$\displaystyle n$ has at most one solution

4)Show that $\displaystyle Ax=b$ where $\displaystyle n=m$ and has rank$\displaystyle n$ has precisely one solution

I'm not sure how to start on either method with 1)

For 2) doing the RRE form method, can be reduced down to a matrix with no nonzero rows, but I get stuck there.

I'm confused about 3, if has rank$\displaystyle n$ does that mean that $\displaystyle m\geq n$?

4) I understand the RRE form method because it can be reduced to the identity matrix, but I don't know how to prove it using the rank nullity formula

Any help would be very much appreciated!