Eigenspace of the transformation

In summary, the conversation discusses finding the eigenvalue and eigenspace of a transformation on R2 that reflects points across a line through the origin. The eigenvalue could be -1 or 1 and the eigenspace is composed of all eigenvectors with that eigenvalue plus the zero vector. The basis of the eigenspace depends on the choice of eigenvectors.
  • #1
fk378
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Homework Statement


Without writing A, find the eigenvalue of A and describe the eigenspace.

T is the transformation on R2 that reflects points across some line through the origin.




The Attempt at a Solution



The eigenvalue could either be -1 or 1. I'm not sure how to figure out the eigenspace of each of these eigenvalues though. And, just to be clear, is the basis of the eigenspace composed of the eigenvectors?
 
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  • #2
The eigenspace of an eigenvalue is the set of all eigenvectors with that eigenvalue plus the zero vector.

fk378 said:
And, just to be clear, is the basis of the eigenspace composed of the eigenvectors?

Depends on which eigenvectors you choose. For example, say you have the matrix

[tex] \[ \left( \begin{array}{ccc} \lambda & 0\\ 0 & \lambda\end{array} \right)\] [/tex]

This matrix has eigenvalue [tex] \lambda [/tex]. Every vector in the plane is an eigenvector of this matrix. If you choose two eigenvectors lying on the same line, they obviously don't span the eigenspace.
 

1. What is the eigenspace of a transformation?

The eigenspace of a transformation is the set of all vectors that are mapped to a scalar multiple of themselves by the transformation. In other words, it is the set of all vectors that remain on the same line or plane after being transformed.

2. Why is the eigenspace important in linear algebra?

The eigenspace is important because it provides insight into the behavior of a transformation on different vectors. It allows for the identification of special vectors that are only scaled by the transformation, rather than being completely transformed into a different direction.

3. How is the eigenspace related to eigenvalues?

The eigenspace is closely related to eigenvalues, as the eigenvalues of a transformation are the scalar values that correspond to the vectors in the eigenspace. Each eigenvalue has its own eigenspace, and the dimension of the eigenspace is equal to the multiplicity of the eigenvalue.

4. Can the eigenspace be empty?

Yes, it is possible for the eigenspace to be empty. This occurs when there are no vectors that are mapped to a scalar multiple of themselves by the transformation. In other words, there are no special vectors that are only scaled by the transformation.

5. How can the eigenspace be calculated?

The eigenspace can be calculated by finding the null space of the matrix A-λI, where A is the transformation matrix and λ is the eigenvalue. The vectors in the null space are the eigenvectors corresponding to that eigenvalue, and the eigenspace is the span of those eigenvectors.

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