T-Invariant Subspaces: Proving W is T-Invariant for E_{\lambda}

[hitmeoff]

Homework Statement

Show that W is a T-invariant subspace of T for:
W = E_{\lambda}

Homework Equations

The Attempt at a Solution

Ok, so I know that I need to show that T maps every element in E_{\lambda} to .

E_{\lambda} = N(T-\lambdaI)

so T must map every eigenvector related to \lambda to another eigenvector in E_{\lambda}

T(x) maps to zero vector, when x is an eigenvector associated with \lambda which is in the eigenspace of \lambda, correct?

 

[Office_Shredder]

Your post is just a huge blank space where you messed up the tex for making a lambda. Can you edit it?

 

[hitmeoff]

fixed

 

[Office_Shredder]

Ok I get it now. E^{\lambda} is the set of all eigenvectors with eigenvalue \lambda. Let's say v is an eigenvector, and Tv=\lambda v. What is T(\lambda v) and how does this help you answer the question?

 

[hitmeoff]

Ok I get it now. E^{\lambda} is the set of all eigenvectors with eigenvalue \lambda. Let's say v is an eigenvector, and Tv=\lambda v. What is T(\lambda v) and how does this help you answer the question?

T(\lambdav) = \lambda²v

and this is just a multiple of T, so T(v) = \lambdav maps to the eigenspace?

 

[hitmeoff]

and another question...

Show that W is a T-invariant subspace when

W = N(T)

So N(T) : { x E W: T(x) = 0} , since W is a subspace it contains the zero vector, thus any vector in N(T) will map to zero which is in W?