Assignment 5 Note that there are only two problems to submit. (1) Write down a linear map S : P2 - P1. Choose a basis for each space and consider the dual bases for : P2 and Pi. Compute the matrices for S and S' by hand and verify that they are transposes of each other. This is an assigned problem, but you do not have to turn in a written solution. (2) Suppose that U, V are subspaces of W. Show that (U + V) = Jon Vo. (3) The double dual space of V, denoted V", is defined to be the dual space of V'. (In other words, V" = (V')'.) Define F : V- V", (Fv)($) = $(v), for v E V and be V. (a) Show that F is a linear map. (b) Show that if T : V - V is linear, than T" . F = FoT, where T" = (I')'. (c) Show that if V is finite dimensional, then F is an isomorphism. Commentary: The interesting feature of this isomorphism is that in contrast to previous proofs that V = V', this isomor- phisms doesn't depend on a choice of basis