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