Question

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.

Answer 1

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.