I am trying to solve the following question:
Let $A$ be a $m\times n$ matrix with rank $r$. If the linear system $Ax = b$ has a solution for all $b\in \mathbb{R}^m$ then I have to check that which of the following statements are true
$(1)$ $ m = r$
$(2)$ The column space of $A$ is a proper subspace of $\mathbb{R}^m$ $(3)$ The null space of $A$ is a non-trivial subspace of $\mathbb{R}^n$, whenever $m = n$. $(4)$ $m \geq n$ implies $m = n$
My approach: If the linear system $Ax = b$ has a solution for all $b\in \mathbb{R}^m$ then for any choice of $b$ there is $x$ such that $Ax = b$ then corresponding linear transformation $T$ must be onto hence rank $T = dim \mathbb{R}^m = m$. Hence option $1$ is true. Further $rank A = m$ implies that $n \geq m$ and if $m \geq n$ then $m = n$ hence option $(4)$ is true. However I am not sure about options $2$ and $3$. Kindly help me out in solving this. If there is any other approach for example counter examples to discard the options then kindly suggest me.
Thank you so much