Linear Algebra Eigenspace Question

[Nexttime35]

Homework Statement

Let T: C∞(R)→C∞(R) be given by T(f) = f'''' where T sends a function to the fourth derivative.

a) Find a basis for the 0-eigenspace.
b) Find a basis for the 1-eigenspace.

The Attempt at a Solution

I just want to verify my thought process for this problem. For a), finding the basis for the 0-eigenspace, essentially I needed to find a basis for the vectors v in V such that T(v) = 0v .

So, would the basis for this 0-eigenspace be all polynomials in P3? If you solve the fourth derivative of any polynomial in P3, you will get 0.

As for b), when finding the basis for the 1-eigenspace, we need to find a basis for the vectors v in V such that T(v) = 1v, or that after solving the fourth derivative, you get a function that is equal to 1? Is this the correct logic? So would the basis for the 1-eigenspace be any polynomial in P4?

Thanks much for your help.

 

[PeroK]

The 1 eigenspace is vectors that map to themselves. Not to the vector 1.

 

[kduna]

If by $C^∞$ you mean the space of all infinitely differentiable functions, then there are a lot more than polynomials around.

Let $f \in C^∞$. Look at the power series: $f(x) = ∑_{i=0}^{∞} a_i x^i$. If the fourth derivative of $f$ is 0, then you have that $a_i = 0$ for $i \geq 4$. Thus the choice of $a_0, a_1, a_2, a_3$ determines $f$ in the 0-eigenspace. Using this, can you come up with a basis? (It will have 4 functions in it).