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.
Show that if {v1,,,,,,,,,,,vp} is linearly dependent in V , then the set of images, {T (v1),,,,,,,,,,T (vp)}, is linearly dependent in W . This fact shows that if a linear transformation maps a set {v1,,,,,,,,,vp} onto a linearly independent set {T (v1),,,,,,,, T (vp)}, then the original set is linearly independent, too (because it cannot be linearly dependent).