- #1
Lily@pie
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Homework Statement
V is a vector space consisting all functions f:R->R that is differentiable many times
(a) Let T:V->V be the transformation T(f)=f'
Find the (real) eigenvectors and eigenvalues of T
(b) Let T be transformation T(f)=f"
Prove that all real number, m is the eigenvalue of T
Homework Equations
For part a, it means to describe the eigenspace of T for each eigenvalue m
The Attempt at a Solution
(a) This means that to find all functions such that T(f)=f'=mf
I separate it into 2 cases, 1st is when m=0
when m=0, all constant function will form the eigenspace of T.
2nd, m is all real number except 0
I can only think of 1 type of function that satisfy f'=mf
which is when f(x)=ae^(bx+c) +d for all real a,b,c,d where a,b=/0 which corresponds to eigenvalue m=ab
However, it seems very not convincing and there seems to be a better way of writing this.
(b) This means that we need to show there exist a eigenspace for all m.
I separated into 3 cases
1st, m=0. All functions of the form f(x)=ax+b for all real a,b where a=/0 is the eigenvector.
2nd, m=-ve. All functions of the form f(x)=a sin(x)+b cos(x) for all non zero a,b.
3rd, m=+ve. I'm clueless ... T-T