Theorem

Let T be a linear map from U to V. Then T is injective if and only if null T = {0}.

${latex.inline\rightarrow}

Suppose T is injective. We already know that ${latex.inline{0} \subseteq null\ T} by 3.10. We need to prove inclusion in the other direction, namely that ${latex.inlinenull\ T \subseteq {0}}. Suppose ${latex.inlinev \in null\ T}. Then we know ${latex.inlineTv = 0 = T(0)}. But we know T is injective, which means that this implies v = 0, and we are done.

${latex.inline\leftarrow}

Suppose null T = {0}. We want to prove that T is injective. Suppose ${latex.inlineu,v \in V} and Tu = Tv. Then, because T is a linear map, ${latex.inlineTu - Tv = T(u - v) = 0}. But this means ${latex.inlineu - v \in null\ T} which means ${latex.inlineu - v = 0} which implies that ${latex.inlineu = v}. This shows that T is injective.