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# Eigenvectors, eigenvalues of a transformation

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Suppose a linear transformation is defined by [see the attachement for the equation] where x is a vector in the euclidean plane (R^2). Answer the following:
a) What are the eigenvalues and eigenvectors of this transformation?
b) Show that the image of x can be represented as a sum of the eigenvectors.
c) What does this transformation represent?

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## SOLUTION This solution is FREE courtesy of BrainMass!

From the condition, we have , where .
(a) We set
So the transform has two eigenvalues 1 and -1.
When the eigenvalue is , the eigenvector is
When the eigenvalue is , the eigenvector is
(b) Consider , then we have

Then we get ,
So the image of is a linear combination of two eigenvectors.
(c) This transform maps to , maps to . Let , then the transform is actually a rotation of , followed by a reflection of the y-axis.

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

Â© BrainMass Inc. brainmass.com December 24, 2021, 10:22 pm ad1c9bdddf>
https://brainmass.com/math/linear-algebra/eigenvectors-eigenvalues-transformation-469484