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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.$T$ is the transformation on $\mathbb{R}^{3}$ that rotates points about some line through the origin.

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Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 1

Eigenvectors and Eigenvalues

Vectors

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allowed one. So this given question, Let's suppose. And let's suppose the elevator access off rotation, like any erected on this line, will not be changed by the rotation means anyway. Active it just inside. And we can say that we have t V equals Toby. Okay, so the line l is in agony space corresponding toe the Eigen value lambda equals to one. Moreover, since rotation does not change land off the vector, there may be only one additional Really Eigen value, which is Lambda. It goes to minus one. If it exists, it means that what? That after the rotation vector, we becomes minus we correct If Lambda is minus one, if this one is existing, it means that after the rotation, the vector we becomes minus we It happens only for the rotation by 1 80 degree, which is perpendicular to the xsl. So in the case, off flotation by one or two degree days additional agony space which is the plane perpendicular to well and the passing through the origin Correct. So this Hagane space corresponds to the Eigen value. Lambda equals two minus one so we can conclude our digital says if the rotation is by Alfa, which is not equals to 1 80 degree. Then there is only one rial Eigen value, which is slammed. Request to one and I gonna space is the excess off rotation. Second conclusion is, if the rotation is Alfa, it was to buy one a two degree. Then there is additional real, like, really Eigen Value, which is Rikers to Lambert, goes to minus one, and it's agony spaces, the plane passing through the origin and perpendicular to the axis off rotation.

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