Prove that the only endomorphism f such that any hyperplane is invariant by it ( f(H) C H) is a homothety.

Question

For the inverse implication it's obvious that any homothety makes any hyperplane invariant. For the direct one , i took a non null linear form φ on a vector space E , its kernel is an Hyperplane H. There exists a vector d in E such that H and K.d are additional in E.Knowing that H is invariant by f, i managed to prove that there exists a scalar λ such that for every x in E we have f(x)-λ.x is in the kernel of φ.The result remains true for every φ in E* because its kernel would be always an hyperplan and then we can prove the existence of another λ. then I wanted to show that for some scalar λ , we have f(x)-λ.x is in the intersection of all kernels of all linear forms which is the trivial space and then conclude that f is an homothety , but the problem remains in the fact that the existence of the scalars depend on the choice of the linear form. Now , I'm asking if this proof can be improved based on that reasoning.If not, suggest me another proof. Thank you.