Got the following problem:
Let $F:R^3 \rightarrow R^3$ a linear transformation so that $F(0,0,1) = (2,-1,\alpha)$
$(1,0,-1)$ and $(0,1,0)$ are eigenvectors of F with associated eigenvalues $3$ and $-9$ respectively. Find $\alpha$ knowing that $-2$ is eigenvalue.
What I guess is that I have to find the eigenvector's basis first, then find $(0,0,1)$ and $(2,-1,\alpha)$ coordinates in that basis, but I can't figure out how to find the third eigenvector that I need to have that basis.
Any hint will be appreciated. Thanks in advance for your time!