Suppose T:VW is a linear transformation and (+) is an inner product on W. (T(v), T(v))w is an inner product on V. (a) If T is one-to-one, show that (V, V2) (b) Show that if A is an invertible nxn real matrix, then the pairing (v.w) = v(ATA)w is an inner product on R". [Hint: Use v (ATA)w = Av Aw and (a).] (e) Is the map defined in part (a) necessarily an inner product if the assumption that T is one-to-one is dropped? Explain why or why not.