Math 308 B: Essay Homework #3 Due: Friday, May 1 in class Problem 1: (a) Suppose that T : Rm Rn is a linear transformation which is one-to-one and suppose that {v1 , v2 } is a linearly independent set of vectors in Rm . Prove that {T (v1 ), T (v2 )} is a linearly independent set of vectors in Rn . (b) Show by example that the conclusion from part (a) can fail if the hypothesis that T be one-to-one is removed. That is, give an example of a linearly independent set of vectors {v1 , v2 } in Rm and a linear transformation T : Rm Rn such that the set {T (v1 ), T (v2 )} is not linearly independent. Problem 2: Let P(2) denote the set of all polynomials of degree at most 2 with real number coecients: P(2) = a0 + a1 t + a2 t2 : a0 , a1 , a2 R Note that if we add two such polynomials, we end up with another polyomial in P(2). Similarly, the set P(2) is closed under scalar multiplication. We can identify P(2) with R3 by sending the polynomial with coecients ai to the vector with entries ai : a0 f (t) = a0 + a1 t + a2 t2 xf = a1 a2 Note that this identication is linear: for any polynomials f (t) = a0 + a1 t + a2 t2 and g(t) = b0 + b1 t + b2 t2 in P(2) and any scalar k in R, we have a0 b0 a0 + b0 a0 ka0 xf + xg = a1 + b1 = a1 + b1 = xf +g and k xf = k a1 = ka1 = xkf a2 b2 a2 + b2 a2 ka2 Dene a function D : R3 R3 by D(xf ) = xf where f denotes the derivative of f . In coordinates, x1 x2 D x2 = 2x3 x3 0 Recall that dierentiation is a linear operator: df dg d (f (t) + g(t)) = (t) + (t) dt dt dt and d df (k f (t)) = k (t) dt dt Because dierentiation is a linear operator and our identication of P(2) with R3 is linear, D is a linear transformation. (Alternatively, one can use the formula for D to directly check D is a linear transformation.) (a) By Theorem 3.8, there is a matrix A such that D(x) = Ax for all x in R3 . Find this matrix. (b) Is D one-to-one? Justify your answer. (c) Is D onto? Justify your answer. 1