Eigenspace of the transformation

[fk378]

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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The eigenspace of an eigenvalue is the set of all eigenvectors with that eigenvalue plus the zero vector.

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

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

This matrix has eigenvalue \lambda. 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.