Thread: Linear Maps

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.


Part 1:2

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

Originally Posted by zangestu888

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.


Part 1:2

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

since this question has too many parts and you've shown 0 work, i'll help you with first two parts only. the kernel of T is the set of all functions f such that T(f) = 0, i.e. f(-x) = -f(x), for all x.

that means the kernel is the set of all odd functions. the image is the set of all even functions: first it's clear that T(f) is even for every function f. conversely if f is even, then T(f/2) = f.

we know that the kernel and image of a linear transformation are subspaces. thus the set of even and odd functions must be subspaces.