$A=(a_{ij})_{m\times n}$ real matrix, $n>m$ then I need to say which of the following are correct statements.

Question

$A=(a_{ij})_{m\times n}$ real matrix, $n>m$ then I need to say which of the following are correct statements.

1. $Ax=0$ has a solution

2. $Ax=0$ has no nonzero solution

3. $Ax=0$ has a nonzero solu

4. dimension of the space of all solutions is atleast $n-m$

I consider $T:\mathbb{R}^n\to \mathbb{R}^m, T(x)=Ax$

$\ker T=\{x: Ax=0\}$ must have a a solution other than $0$. Suppose only $1$ is true i.e $\dim \ker T=0$, then $\dim\ker T+\dim Im T=n\Rightarrow\dim ImT=n>m\Leftrightarrow$

If $\dim imT=m$ then the dimension of solution space will be atmost $n-m$ so 1,3,4 are correct?

Answer 1

Yes: your solution is correct.