Injective

Let $$V_1$$ and $$V_2$$ be normed spaces and let $$T$$ be an operator $$T:V_1\to V_2$$. Then, $$T$$ is injective(see Def 2.4.5 ) if $$T\mathbf{v} = \mathbf{0} \implies \mathbf{v} = \mathbf{0}.$$

If $$T$$ is a matrix, this means that it has full column rank, because the definition above suggests that the nullspace of $$T$$ is of dimension 0, which, according to the fundamental theorem of linear algebra, suggests that $$T$$ has to have full column rank. An injective matrix $$T$$ is also called one-to-one.