Let be a basis for a vector space V. Prove that if a linear transformation satisfies then T is the zero transformation.
Getting Started: To prove that T is the zero transformation, you need to show that for every vector v in V.
(i) Let v be an arbitrary vector in V such that
(ii) Use the definition and properties of linear transformations to rewrite as a linear combination of .
(iii) Use the fact that to conclude that , making T the zero tranformation.