(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....

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. "/>
Extracted 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.