Eigenvectors of a Symmetric Endomorphism

Question

Prove that there isn't any symmetric endomorphism $f$ of $\mathbb R^3$ such that $e_1=(1,0,1)$ and $e_2=(1,1,1)$ are eigenvectors of $f$.

I don't know how to do it, any hint?

Answer 1

You can't prove it, because it's false.

Eigenvectors relative to distinct eigenvalues of a symmetric matrix are orthogonal, and $e_1$ is not orthogonal to $e_2$. But the two vectors could be relative to the same eigenvalue.

Indeed, there are several symmetric matrices with this property, for instance $$ \begin{bmatrix} 1 & 0 & 1 \\ 0 & 2 & 0 \\ 1 & 0 & 1 \end{bmatrix} $$