MATH 307 Assignment 5Due Wednesday, April 1, 2015 in classDo each of the following questions from the exercises at the end of each section of thetextbook.You must show your work to get full marks. For some questions, I have givenhints, clarifications, or extra instructions.Be sure to follow these!1)9.1#162)9.1#52. You are essentially being asked to prove that ifT:V→Wis anisomorphism, and!v1,…,!vm{}is a basis forV, thenT(!v1),…,T(!vm){}is abasis forW. (And soVandWhave the same dimension.) That is, you needto show that this set is independent and spansW3)9.2#48. [Hint: Theorem 9.7]4)9.3#22. Instructions are before exercise #19. Also state clearly what thevector spacesVandWare..5)9.4#146)9.4#387)“Calculus required”The notationC∞(R)is used for the subspace ofC(R)consisting of allfunctionsf:R→Rthat are infinitely differentiable, i.e. whosederivativesf(k)(x)exist for allk≥1. For example, all polynomialfunctions are inC∞(R)Note (from calculus):•C∞(R)is a subset ofC(R)because all differentiable functions are.