Let V be a ﬁnite-dimensional vector space, and let λ be any scalar. For any ordered basis β for V , prove that [ λ I V ] β = λI . Compute the characteristic polynomial of λ I V . Show that λ I V is diagonalizable and has only one eigenvalue.

Let V be a ﬁnite-dimensional vector space, and let λ be any scalar. For any ordered basis β for V , prove that [ λ I V ] β = λI . Compute the characteristic polynomial of λ I V . Show that λ I V is diagonalizable and has only one eigenvalue.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let V be a ﬁnite-dimensional vector space, and let λ be any scalar.

For any ordered basis β for V , prove that [ λ I V ] β = λI .

Compute the characteristic polynomial of λ I V .

Show that λ I V is diagonalizable and has only one eigenvalue.

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