How can i do this:
given f:R---->R is called even function f(-x)=f(x) respectively f(-x)=-f(x) for all x in real. Let E be set of even functins in F[R] and let O be set of odd functions in F[R].
(1)Let T:F[R]---->F[R] be defined by assigning to the function f:R--->R the function T(f) defined by (t(f)))(x)=f(x)+f(-x).This is a linear map, find the kernal and the image of T.
Prove that E and O are subspaces of F[R] ( hint given you can use the above relation??))
Fix a positive nartual number n.Find a basis of teh subspace even polynomials in Pn, determin its dimension
Determine the dimension of the subspace of odd polynomials in Pn