Let u be an eigenvector of A corresponding to an

Chapter , Problem 11E

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QUESTION:

Let u be an eigenvector of A corresponding to an eigenvalue \(\lambda\), and let H be the line in \(\mathbb{R}^n\) through u and the origin.

a. Explain why H is invariant under A in the sense that Ax is in H whenever x is in H.

b. Let K be a one-dimensional subspace of \(\mathbb{R}^n\)  that is invariant under A. Explain why K contains an eigenvector of A.

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QUESTION:

Let u be an eigenvector of A corresponding to an eigenvalue \(\lambda\), and let H be the line in \(\mathbb{R}^n\) through u and the origin.

a. Explain why H is invariant under A in the sense that Ax is in H whenever x is in H.

b. Let K be a one-dimensional subspace of \(\mathbb{R}^n\)  that is invariant under A. Explain why K contains an eigenvector of A.

ANSWER:

Solution 11E(a), , we have , so by the definition o

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