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.

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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