Guided Proof Prove that T: VW is a lineartransformation if

Chapter 6, Problem 6.1.84

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Guided Proof Prove that T: VW is a lineartransformation if and only ifT(au + bv) = aT(u) + bT(v)for all vectors u and v and all scalars a and b. Getting Started: This is an if and only if statement,so you need to prove the statement in both directions.To prove that T is a linear transformation, you needto show that the function satisfies the definition ofa linear transformation. In the other direction, let Tbe a linear transformation. Use the definition andproperties of a linear transformation to prove thatT(au + bv) = aT(u) + bT(v). (i) Let T(au + bv) = aT(u) + bT(v). Show that Tpreserves the properties of vector addition andscalar multiplication by choosing appropriatevalues of a and b.(ii) To prove the statement in the other direction,assume that T is a linear transformation. Use theproperties and definition of a linear transformationto show that T(au + bv) = aT(u) + bT(v)..