Find a basis S in $R^{4}$ such that $M_{S}(F)$ is diagonal

Question

$F(x,y,z,t) = (5x+6y+3z-6t, 2y, -6x-6y-4z+6t, 3y-t)$. Find a basis $S$ in $R^{4}$ such that that $M_{S}(F)$ is diagonal. Find also the matrix itself.

I have never met task like this one and I don't really don't know what should be my first step.

I tried to find a basis for $(5x+6y+3z-6t, 2y, -6x-6y-4z+6t, 3y-t)$ and I got an identity matrix. However I don't really fell if this step even helped something.

Any tips?

Answer 1

Simply find a basis of eigenvectors of this endomorphism. That's all.