RHETOR 103A Chapter 12: R103a Reading Plato Apology

If Tj : P(R) rightarrow P(R) be the linear operator sending a polynomial to its j-th derivative. Prove that T1,... ,Tn are linearly independent in L(P(R)). Let T : V rightarrow W be a linear transformation between F-vector space V and W of equal dimension. Prove that there exist basis beta of V and gamma of W such that [T]gamma beta is a diagonal matrix whose entries are 1 or 0.

60. Suppose we know that u v-u w. Does it follow that v d in R" w?Ifit does, give a proof that is vali otherwise, give a counterexample (i.e., a specific set of vectors u, v, and w for which u v-u w but v + w)