Let $c_{1}.....c_{n}$ be scalars, not all zero, such that $\sum_{i=1}^{n}c_{i}a_{i}$ = 0 where $a_{i}$ are column vectors in $R^{n}$.
Consider the set of linear equations
$Ax = b$
where $A=\left [ a_{1}.....a_{n} \right ]$ and $b=\sum_{i=1}^{n}a_{i}$. The set of equations has
- a unique solution at $x=J_{n}$ where $J_{n}$ denotes a $n$-dimensional vector of all 1.
- no solution
- infinitely many solutions
- finitely many solutions
