## anonymous 5 years ago Show that a linear transformation T:V-->W is injective if and only if it has the property of mapping linearly independent subsets of V to linearly independent subsets of W.

Let $$x\in \ker T$$. If $$x\neq 0$$ then $$\{x\}$$ is linearly independent. But $$\{T(x)\}=\{0\}$$ is linearly dependent. Hence $$x=0$$ and therefore $$T$$ is injective.