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(a) Let V be a finite dimensional vector space and T E L(V). Suppose that the rank of T is 1. Prove that every nonzero vector in the range of T is an eigenvector of Т. (b) Let T E L(C") where n > 1. Suppose that the matrix of T with respect to the standard basis of C" is 1 1 1 1 1 ... 1 ... 1 1 1 ... Find, with explanation, the characteristic polynomial and the minimal polyno- mial of T.

(a) Let V be a finite dimensional vector space and T E L(V). Suppose that the rank of T is 1. Prove that every nonzero vector in the range of T is an eigenvector of Т. (b) Let T E L(C") where n > 1. Suppose that the matrix of T with respect to the standard basis of C" is 1 1 1 1 1 ... 1 ... 1 1 1 ... Find, with explanation, the characteristic polynomial and the minimal polyno- mial of T.

Advanced Engineering Mathematics
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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(a) Let V be a finite dimensional vector space and T E L(V). Suppose that the rank of T is 1. Prove that every nonzero vector in the range of T is an eigenvector of Т. (b) Let T E L(C") where n > 1. Suppose that the matrix of T with respect to the standard basis of C" is 1 1 1 1 1 ... 1 ... 1 1 1 ... Find, with explanation, the characteristic polynomial and the minimal polyno- mial of T.
Transcribed Image Text:(a) Let V be a finite dimensional vector space and T E L(V). Suppose that the rank of T is 1. Prove that every nonzero vector in the range of T is an eigenvector of Т. (b) Let T E L(C") where n > 1. Suppose that the matrix of T with respect to the standard basis of C" is 1 1 1 1 1 ... 1 ... 1 1 1 ... Find, with explanation, the characteristic polynomial and the minimal polyno- mial of T.
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