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Chapter 5: Q31E (page 267)

Question: In Exercises 31 and 32, let A be the matrix of the linear transformation T. Without writing A, find an eigenvalue of A and describe the eigenspace.

31. T is the transformation on \({\mathbb{R}^2}\) that reflects points across some line through origin.

Short Answer

Expert verified

The eigenvalue is \(1\) , and the eigenspace is \({\rm{span}}\left\{ x \right\}\).

Step by step solution

01

Given information

Consider \(T\) be the transformation on \({\mathbb{R}^2}\) and \(T\) reflects points across the line through the origin.

02

Find eigenvalue and eigenspace of A

Consider A be the matrix of the linear transformation T.

The equation of the transformation matrix is as follows:

\(T = Ax\) 鈥� (1)

And, the transformation matrix reflects the lines through the origin. So, the equation will become:

\(T = x\) 鈥� (2)

From (1) and (2), we have

\(Ax = x\)

That implies \(x\) is an eigenvector of \(A\) corresponding to the eigenvalue of \(1\).

Therefore, the eigenspace is Span \(\left\{ x \right\}\).

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