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Let $A\in M_{m\times n}(\mathbb{F})$. Consider the map

\begin{aligned} T_A:\mathbb{F}_c^n&\to \mathbb{F}_c^m\\ X&\mapsto AX \end{aligned}

I want to show that $T_A$ is a surjection if and only if the rank of $A$ is $m$. Could someone enlighten me how to start?

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    That depends: what's your definition of rank? – 2017-01-30

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Hint: $x \mapsto Ax$ is surjective if and only if every equation of the form $Ax = b$ (for some $b \in \Bbb F^m$) has a solution.