- #1
uiulic
- 99
- 0
Homework Statement
This is a problem related to linear map over vector spaces of functions and finding kernels.
Let V be the vector space of functions which have derivatives of all orders, and let D:V->V be the derivative. Problem1: What is the kernal of D?
Problem2: Let L=D-I,where I is the identity map of V. What is the kernel of L?
Homework Equations
problem1: let f be an unknown element of the kernel, then D(f)=0
problem2: let g be an unknown element of its kernel, then L(g)=0
Note:The above 0 is a 0 function.
The Attempt at a Solution
My Solving problem1: D(f)=0, but the unknown is a function. I don't have a clue to solve such "equations" (with the unknowns of functions). So I can only guess the answer.
My Solving problem2: L(g)=0 => (D-I)(g)=0 =>D(g)-I(g)=D(g)-g=0 noting D-I must also be a linear map. So I need to solve D(g)=g. Since g is a function, again I can only guess the answer.
There is a problem with my "guessed" answer, because I can only prove my "guessed" answer (is a set) meet the equation, but I have no idea whether there exists a solution which does not lie in my guessed set.
Thanks