What is the dimension of eigenspaces in a function-based linear transformation?

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In summary, the linear transformation f maps functions in the space C^∞(R) to their derivatives. To solve for eigenvalues, one must set up and solve the eigenvalue problem. In this case, the solutions are of the form ke^λt. When λ is equal to -5 or 0, the corresponding eigenspaces are one-dimensional, as every solution can be written as a constant multiple of e^{\lambda x}.
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Tala.S
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Linear transformation f:C^∞(R) -> C^∞(R)

f(x(t)) = x'(t) a) I have to set up the eigenvalue-problem and solve it :

My solution : ke^λtb) Now I have to find the dimension of the single eigen spaces when λ is

-5 and 0. My solution :

Eigenspaces :

E-5 = ke^-5t

E0=k (because ke^0t = k)

But I don't know how to find the dimension of the single eigen spaces ?

I'm used to working with vectors but now it's functions and I'm not sure about the dimension.
 
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Solved !
 
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Good! I presume that you realized that since every solution, [itex]Ce^{\lambda x}[/itex] is a constant, C, times the single function [itex]e^{\lambda x}[/itex], the space is one dimensional.
 

1. What is an eigenspace?

An eigenspace is a vector space associated with a given eigenvalue of a linear transformation. It is the set of all eigenvectors corresponding to that eigenvalue, along with the zero vector.

2. How is an eigenspace related to the concept of eigenvalues?

An eigenspace is directly related to eigenvalues, as it is the set of all vectors that are transformed only by scalar multiplication when the linear transformation is applied. This scalar value is the corresponding eigenvalue.

3. Can an eigenspace have more than one eigenvalue?

Yes, an eigenspace can have multiple eigenvalues. This is because a linear transformation can have more than one eigenvector that corresponds to different eigenvalues.

4. How does the dimension of an eigenspace relate to the dimension of the original vector space?

The dimension of an eigenspace is always less than or equal to the dimension of the original vector space. This is because an eigenspace is a subspace of the original vector space and therefore cannot have a higher dimension.

5. How can eigenspaces be used in practical applications?

Eigenspaces are commonly used in data analysis and machine learning, particularly in the field of principal component analysis. They can also be used in solving systems of linear equations and in understanding the behavior of dynamic systems in physics and engineering.

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