Determine whether the following are linear transformations from R2 into R2. In each case, if the map is a linear transformation, provide a proof that it is a linear transformation by verifying that (LT1) and (LT2) hold. If the map is not a linear transformation, give a counterexample to show (LT1) or (LT2) fails, or show the 0 test fails (see Q5 in Linear Transformations for the 0 test)). (a) T 4x - 2y 7x+ 3y ( b ) I (C) T ([= ]) - 123 (d) Let w be a fixed, non-zero vector in R2. For every u E R2, T(u) = 2u - w . uwDefinition: A transformation T : Rn -Pom is said to be a linear transformation if the following conditions hold: (LT1) T(utv) = T(u) + T(v) for all u, v E R", and (LT2) T(cv) = cT(v) for all v E R" and all scalars CERFact: Let 0, be the zero vector in R" and 0, be the zero vector in R" . If T : R" -> R is a linear transformation, then T(0 = 0 m that is, a linear transformation sends the zero vector of the domain to the zero vector of the codomain