Question

5. consider set F(R):ff: f:R-R), but set all function with set real number in domain and codomain. Show "addition" in any two function it.eCE(R) to produce new function such as given: ttgR2R which is every xER such as given:(tg)lx)-fx)+g(x), and any real number k ER, multiply it with any element f EF(R) to produce new function as given: kfRR in every value xER such as given:(k:0(x):-kfx)(observe it with multiply dua real number) (a) Show. FIR) ith addition and multiplicarion operation on real number vector space. (b) What is zero vector ine(R)? (c) Find the inverse vactor for vector f(x)--7e-sinx in F(R). (d) Consider subset from F(R) such as given: E(R):={feF(R): f for even function ii. f(x)=f(-x)VxER) Show E(R) is a vector subspace from F(R). [12 marks]

help me.

Answer 1

page(リ addition:-++g:R--R defined as Scalea multiplication'- af: R→R delined-a4 Where x EIR c lain = font (a)+h(m) Associctivity hold's Cmstan O : Such that (f +0)(n)=f(x)+0(1): (04)) So o ,s additive i R be So,-土[R_HR also function.

рад е② HeneefEFRuch thot Exeay element in FíR) has addlitihe iovepte Comsiderr vii) Y111

Consider ex) Hence Fr) is vectox space bven cleare zero.pvector lef-E(R)ǐ5 a C funatiom ocolich eo: RR detined as cens vecto f FCR) of(20t Hence olf +pa EE(R) B1 subspace ritema