You should clearly define or describe all terms (e.g. $\theta$, $\beta^*$, etc.) in order to receive assistance. Problems of contact geometry require...
If $Ax = b$ is solvable for every $b\in \Bbb R_m$, then every vector in $\Bbb R_m$ is an element of the column space of $A$. Hence, $\Bbb R_m$ equals...