Differentiable Linear Transformation

[Needhelpzzz]

Homework Statement

Let V be the linear space of all real functions Differentiable on (0,1). If f is in V define g=T(f) to mean that g(t)=tf'(t) for all t in (0,1). Prove that every real λ is an eigenvalue for T, and determine the eigenfunctions corresponding to λ.

Homework Equations

The Attempt at a Solution

All I know is that f'=λf and T(f)=λf in general. I tried substituting the variables, and I ended up with only t, which doesn't make sense.

 

[Fredrik]

Can you use the definitions of "eigenvalue", "eigenfunction" and T to explain what "f is an eigenfunction of T with eigenvalue λ" means?

Edit: You have already given a partial answer for that by saying that Tf=λf. This is the part that follows from the definitions of eigenvalue and eigenfunction. So now you need to use the definition of T to explain what Tf=λf means.