Prove that if dimV=1\dim V = 1dimV=1 and T∈L(V,V)T \in \mathcal L(V,V)T∈L(V,V) then there exists λ∈F\lambda \in Fλ∈F such that T(v)=λvT(v) = \lambda vT(v)=λv for all v∈Vv \in Vv∈V.
Let vvv be a basis for VVV, and let λ\lambdaλ be such that Tv=λvTv = \lambda vTv=λv. Since any w∈Vw \in Vw∈V can be written as w=cvw = cvw=cv for c∈Fc \in \mathbf Fc∈F we have