Question: 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,,,,,,,,,,,vp} 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 (vp)} is linearly dependent, then {v1,,,,,,,,vp} 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).