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Transformations U = X Y, and V = X − Y (a) (3 points) Derive E(U) and E(V) in terms of µX and µY E(U) = E(XY) = E(X) E(Y) = µX µY E(V) = E(X − Y) = E(X) − E(Y) = µX − µY (b) (5 points) Derive E(U V) in terms of µX, µY, and 2 σX 2 σY Hint you want the expected value of U times V Use the definitions at the top of1 The kernel of T is defined by ker T = {v T(v) = 0} 2 The range of T = {T(v) v is in V} Theorem Let T V 6 W be a linear transformation Then 1 Ker T is a subspace of V and 2 Range T is a subspace of W Proof 1 The kernel of T is not empty since 0 is in ker T by the previ ous theorem Suppose that u and v are in ker T so that T(u) = 0 and T(v) = 0Then T(u v) = T(u) T(v) = 0Can choose the zero at any point 2 ƒƒ"ƒY ƒ{ƒu ƒ}ƒbƒVƒ… ˆá‚¢