# Linear independence of vectors (involving absolute values of components)

If $$\mathbf a_1,...,\mathbf a_n$$ are n vectors in $$\mathbb C^n$$ with $$\mathbf a_i = (a_{i1},...,a_{in})$$ and such that $$\lvert a_{jj} \rvert \gt \sum_{\overset{i=1}{i\ne j}}^{n} \lvert a_{ij} \rvert \space\space \text{for j=1,...,n}$$ how can one show that they are linearly independent?

For n=3, for example, I tried assuming that one of the vectors is a linear combination of the other two and then using the property given above and the usual inequalities ($$\lVert \mathbf x + \mathbf y \rVert \leq \lVert \mathbf x \rVert + \lVert \mathbf y \rVert$$ and $$\lvert \space \lVert \mathbf x \rVert - \lVert \mathbf y \rVert \space \rvert \leq \lVert \mathbf x + \mathbf y \rVert$$) to show that you would have both $$\lvert \lambda \rvert \lt \lvert \mu \rvert$$ and $$\lvert \mu \rvert \lt \lvert \lambda \rvert$$ if, say, $$\mathbf a_1 = \lambda \mathbf a_2 + \mu \mathbf a_3$$, but only got \begin{align} \lvert\lambda\rvert + \lvert\mu\rvert\geq 1\\ (\lvert\lambda\rvert - 1)\cdot \lvert a_{22}\rvert\lt(\lvert\mu\rvert - 1)\cdot \lvert a_{32}\rvert \\ (\lvert\mu\rvert - 1)\cdot \lvert a_{33}\rvert\lt(\lvert\lambda\rvert - 1)\cdot \lvert a_{23}\rvert\end{align}