eigenvector

Suppose T is in L(V) is such that every vector in V is am eigenvector
of T.Prove that T is a scalar multiple of the identity operator.

Proof:- Assume T is in L(V), let T be in V.

Then Tv= Lv for all v in V and Lambda in F

where F is a field

Tv= LIv where I is the identity map.

This is where i am stuck, where do i go from here please?

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