Finding Eigenvalues and Eigenspaces: A Reflection Transformation Example

[fk378]

Homework Statement

Let A be the matrix of the linear transformation T. Without writing A, find an 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

Since they tell us that the point is reflected across the origin, I say that the eigenvalue= -1 and since T is a linear transformation, the eigenspace is in R2.

Are both the answer and reasoning correct?

 

[Dick]

Vectors are differences between points, not the points themselves. In this case you should be thinking of the vector corresponding to (x,y) to be the arrow connecting (0,0) with (x,y). Now consider the two cases of vectors parallel to the line and perpendicular to the line separately.

 

[fk378]

I don't think I understand how to find the eigenvalue when considering the parallel and perpendicular vectors.

 

[Dick]

Take an example, suppose the line is x=y. What are A((1,1)) and A((1,-1))? How did I pick those two vectors?