Finding Eigenvalues and Eigenspaces: A Reflection Transformation Example

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Homework Help Overview

The discussion revolves around finding eigenvalues and eigenspaces related to a linear transformation that reflects points across a line through the origin in R2.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • Participants explore the concept of eigenvalues in the context of reflection transformations, with one participant suggesting an eigenvalue of -1 and discussing the corresponding eigenspace in R2. Others raise questions about the reasoning behind this choice and the implications of considering vectors parallel and perpendicular to the line of reflection.

Discussion Status

The discussion is ongoing, with participants questioning the correctness of the initial reasoning and exploring different cases for vectors related to the transformation. Some guidance has been offered regarding the interpretation of vectors in this context, but no consensus has been reached.

Contextual Notes

Participants are considering specific cases of reflection, such as the line x=y, and discussing how to approach the problem without explicitly writing the transformation matrix.

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

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