= Solution
The coefficient matrix is a <Vandermonde matrix>, with determinant
$$
\det\begin{pmatrix}1&1&1\\a&b&c\\a^2&b^2&c^2\end{pmatrix}=(b-a)(c-a)(c-b).
$$
Therefore a nonzero solution exists exactly when
$$
\boxed{a=b\quad\text{or}\quad a=c\quad\text{or}\quad b=c.}
$$
For 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>.
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