How can I prove that a linear map from a 1-dimensional space to itself is really just multiplication by some scalar?

Question

Show that every linear map from a 1-dimensional vector space to itself is multiplication by some scalar. More precisely, prove that if the dimension of $V$ is equal to 1, and $T \in L(V,V)$, then there exists $\lambda \in F$ such that $T(v) = \lambda v $ for al $v \in V$.

Answer 1

Recall that the transformation is completely determined once we define

$$T(1)=\lambda \implies \forall a\in V \quad T(a)=T(a\cdot 1)=a\cdot T(1)=a \cdot\lambda$$