MAT2611/101/3/2020 ASSIGNMENT 03 Due date: Thursday, 30 April 2020 UNIQUE ASSIGNMENT NUMBER: 647195 ONLY FOR SEMESTER 1 INSTRUCTIONS FOR THE ASSIGNMENT Take care to explain all your arguments. If you choose to submit via my Unisa, note that only PDF files will be accepted. Problem 7. A function R - R is said to be even (respectively, odd) if for each r ER, f(-x) = f(x) (respectively, f(-x) = -f(x)). Let Ueven (respectively, Uodd) denote the set of even (respectively, odd) functions on R. Show that the vector space RE of all real valued functions on R is a direct sum of Ueven and Voda with Ueven NUodd = {0} In other words, prove RIK = Ueven D Uodd [5 marks] Problem 8. Given the polynomials po, Pi, ..., Pm are polynomials in Pm [C] such that p, (2) = 0 for each j = 0, 1, ..., m. Show that (Po, Pi, ..., Pm) is not linearly independent in Pm [C]. What is span po, P1, . .., Pm]? [5 marks] Problem 9. Show that if U1, U2, ..., Um are finite dimenasional subspaces of a vector space V such that : UinUj = (10}, ifif i, Ui, if i = j. Show that: dim (U1 0 U2 0 . . . O Um) = dim U1 + dim U2 + . . . + dim Um Guess and prove a formula for dim (U1 + U2 + . . . + Um). [5+ 5 = 10 marks] [Total: 20 marks] - End of assignment