Minimum distance of a linear code and its relation with a control matrix of the code

Solution 1:

Denote by $d(H)$ the largest integer s.t. any $d(H)-1$ columns in $H$ are linearly independent. You have shown that two integers, $d(\mathcal{C})$ and $d(H)$, are identical, i.e., $d(H) = d(\mathcal{C})$. So you are done, as this is the same as saying $d(H)=d\Longleftrightarrow d(\mathcal{C}) = d$ for some $d$.

There is however one small technical flaw in your argument (which luckily does not affect the proof). You write, "any combinations of $\geq d$ of them, will form a linearly-dependent vector family (this is by hipothesis)}". This is however not true. There might be $d$ columns which are linearly independent, as $d$ is defined largest integer such that any $d-1$ columns are linearly independent.