Theorem

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

Proof

Per 1753751116 - Axler 3.10 Linear maps take 0 to 0|3.10, we know ${latex.inline0 \in range\ T} (since T(0) = 0).

Now suppose \({latex.inline[w_{1}, w_{2} \in range\ T](w_{1}, w_{2} \in range\ T)} . We want to show that \){latex.inlinew{1} + w{2} \in range\ T}. Well, \({latex.inline[w_{1} + w_{2} = Tv_{1} + Tv_{2} = T(v_{1} + v_{2})](w_{1} + w_{2} = Tv_{1} + Tv_{2} = T(v_{1} + v_{2}))}, and we know \){latex.inlinev{1} + v{2} \in V}, so we have what we need.

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