Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2009 / Paper 2 / 14 / c / Solution

The coefficient matrix is a Vandermonde matrix, with determinantTherefore a nonzero solution exists exactly whenFor example if $a=b$, the first two columns are equal and $(1,-1,0)$ is a solution. If exactly two parameters agree, the distinct remaining column is independent of the repeated one, so the kernel has dimension one. If all three agree, only $x+y+z=0$ remains, giving a two-dimensional kernel.