Let $c_1, \ldots, c_n$ be scalars, not all zero, such that $\sum_{i=1}^n c_i a_i = 0$ where $a_i$ are column vectors in $\mathbf{R}^n$. Consider the set of linear equations $$Ax = b$$ where $A = [a_1, \ldots, a_n]$ and $b = \sum_{i=1}^n a_i$. The set of equations has

1. Since there exist scalars $c_1, \ldots, c_n$, not all zero, such that $\sum_{i=1}^n c_i a_i = 0$, the column vectors $a_1, \ldots, a_n$ of the $n \times n$ matrix $A$ are linearly dependent.

2. Linear dependence implies that $\text{rank}(A) < n$, so the homogeneous equation $Ax = 0$ has non-trivial solutions (the nullity of $A$ is at least $1$, with vector $c = [c_1, \ldots, c_n]^T \neq 0$ satisfying $Ac = 0$).

3. The vector $b$ is given by:

   $$b = \sum_{i=1}^n a_i = A J_n$$

   where $J_n = [1, 1, \ldots, 1]^T$. This shows that $x = J_n$ is a particular solution to $Ax = b$, meaning the system is consistent.

4. Since the system $Ax = b$ is consistent and $\text{rank}(A) < n$, it must have infinitely many solutions of the form $x = J_n + k c$ for any scalar $k \in \mathbb{R}$.

Hence, the correct option is (C).