Explanation :
The equation Ax = b becomes according to what is given
$ \left[{\mathrm a}_1,{\mathrm a}_2,...{\mathrm a}_\mathrm n\right]\begin{bmatrix}{\mathrm x}_1\\{\mathrm x}_2\\.\\.\\.\\{\mathrm x}_\mathrm n\end{bmatrix}={\mathrm a}_1+{\mathrm a}_2+{\mathrm a}_3+...{\mathrm a}_\mathrm n $
where a_{i} are column vectors in R^{n} but since we have C_{i} (not all zero) such that.
$ \sum C_\mathit j\alpha_\mathit j\mathit=0 $
it means the n column vectors are not linearly independent and hence
rank (A) < n
So we have infinitely many solutions one of which will be J_{n}, where J_{n} denotes a n-dimensional vector of all 1.