one-to-one

T:V\to W is one-to-one (injective) when distinct inputs always produce distinct outputs: T(\mathbf u)=T(\mathbf v) forces \mathbf u=\mathbf v. No output is ever shared by two different inputs, though an output can be missed entirely. For a linear T, one-to-one is equivalent to \ker T=\{\mathbf 0\}.