Theorem

If T is a linear map, then range T is a subspace of W.

Proof

Per 3.10, we know ${latex.inline0 \in range\ T} (since T(0) = 0).

Now suppose ${latex.inlinew_{1}, w_{2} \in range\ T} . We want to show that ${latex.inlinew_{1} + w_{2} \in range\ T}. Well, ${latex.inlinew_{1} + w_{2} = Tv_{1} + Tv_{2} = T(v_{1} + v_{2})}, and we know ${latex.inlinev_{1} + v_{2} \in V}, so we have what we need.

Now suppose ${latex.inlinew_{1} \in range\ T}. Then ${latex.inlinew_{1} = Tv_{1}} which implies ${latex.inline\lambda w_{1} = \lambda Tv_{1} = T(\lambda v_{1})}. But we know that ${latex.inline\lambda * v_{1} \in V}, so we get what we need because ${latex.inline\lambda * w_{1}} is the result of applying T to ${latex.inline\lambda * v_{1}}.