Eigenvalue of vector space of polynomials

In summary, the conversation discusses the linear map D:V→V defined by D(f)=f', where f' represents the derivative of the polynomial f. It is shown that D11=0, meaning that the 11th derivative of any polynomial with degree less than 10 is equal to zero. This leads to the deduction that 0 is the only eigenvalue of D. To find a basis for the generalized eigenspaces V1(0), V2(0), and V3(0), one can consider the degree of D(f) for a polynomial f in V. This shows that the possibility of a nonzero eigenvalue is eliminated, and therefore 0 is the only eigenvalue. The problem of finding a basis
  • #1
specialnlovin
19
0
Let V=C[x]10 be the fector space of polynomials over C of degree less than 10 and let D:V[tex]\rightarrow[/tex]V be the linear map defined by D(f)=f' where f' denotes the derivatige. Show that D11=0 and deduce that 0 is the only eigenvalue of D. find a basis for the generalized eigenspaces V1(0), V2(0), V3(0)
I get that D11=0 because a polynomial with degree 10 has an 11th derivative equal to zero since. However I am not sure how exactly to write that in a proof, also how to then deduce that the eigenvalue must equal zero.
I also don't know exactly how to start the problem of finding a basis for the eigenspaces.
 
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  • #2
If f is in V, so it's a polynomial of degree n<=10, what's the degree of D(f)? What does that tell you about the possibility of a nonzero eigenvalue?
 

Related to Eigenvalue of vector space of polynomials

1. What is an eigenvalue of a vector space of polynomials?

An eigenvalue of a vector space of polynomials is a scalar value that, when multiplied by a polynomial vector, results in a scalar multiple of that same vector. In other words, it is a special value that represents the scaling factor of a polynomial vector when it is transformed by a linear operator.

2. How is the eigenvalue of a polynomial vector calculated?

The eigenvalue of a polynomial vector can be calculated by solving the characteristic polynomial equation for the given vector space. This equation involves finding the roots of a polynomial, which can be done using various methods such as factoring or using the quadratic formula.

3. Why are eigenvalues important in the study of polynomials?

Eigenvalues are important in the study of polynomials because they provide valuable information about the behavior and transformations of polynomial vectors. They can be used to determine the stability of systems, find critical points, and solve differential equations.

4. Can a polynomial vector have more than one eigenvalue?

Yes, a polynomial vector can have multiple eigenvalues. In fact, most polynomial vectors have multiple eigenvalues, which can be found by solving the characteristic polynomial equation. Each eigenvalue corresponds to a different scaling factor and direction of transformation for the polynomial vector.

5. How are eigenvalues and eigenvectors related in a vector space of polynomials?

In a vector space of polynomials, an eigenvalue and its corresponding eigenvector are directly related. The eigenvector is the actual polynomial that is transformed by the linear operator, while the eigenvalue is the scalar factor that represents the magnitude of the transformation. Together, they form an eigenpair that helps us understand the behavior of polynomial vectors in a given vector space.

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