Reveal an important connection between linear independence and linear transformations and provide practice using the definition of linear dependence. Let V and W be vector spaces, let T: V → W be a linear transformation, and let {v₁,...,v_p} be a subset of V.

Suppose that T is a one-to-one transformation, so that an equation T(u) = T(v) always implies u = v. Show that if the set of images {T(v₁),...,T(v_p)} is linearly dependent, then {v₁,...,v_p} is linearly dependent. This fact shows that a one- to-one linear transformation maps a linearly independent set onto a linearly independent set (because in this case the set of images cannot be linearly dependent).