Let $V$ be a normed vector space and $\{v_k\}_1^N$ a collection of vectors in $V$. Assume that there exists a constant $A>0$ such that the inequality \begin{equation*} A\sum_{k=1}^N |c_k^2| \leq \|\sum_{k=1}^N c_kv_k\|^2 \end{equation*} holds for all scalar coefficients $c_1,\ldots c_N$. Show that the vectors $\{v_k\}_1^N$ are linearly independent.
This is trivial because for hypothesis $\sum_{k=1}^N c_kv_k=0$, so $A\sum_{k=1}^N |c_k^2|\leq 0$. Then, $|c_k|^2=0$ and $c_k=0$ for all $k$. My question is: why the problem suppose that exists a constant $A$?
It seems as though what you're trying to ask is something like the following.
The answer to this is probably that the inclusion of the constant $A$ makes it so that the inequality becomes a necessary and sufficient condition for linear independence. In other words: it is true that if the vectors $v_i$ are independent, then there exists an $A>0$ for which $A\sum_{k=1}^N |c_k^2| \leq \|\sum_{k=1}^N c_kv_k\|^2$. It is not true that we can always take this constant to be $A = 1$.
So, the inclusion of the constant $A$ has nothing to do with the exact question being asked, but likely has to do with the context in which it will be used.