Chapter 4.3, Problem 31E

Exercises 31 and 32 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.

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