Label the following statements as true or false. (a) Every linear operator on an

QUESTION:

Label the following statements as true or false. (a) Every linear operator on an n-dimensional vector space has n distinct eigenvalues. (b) If a real matrix has one eigenvector, then it has an infinite number of eigenvectors. (c) There exists a square matrix with no eigenvectors. (d) Eigenvalues must be nonzero scalars. (e) Any two eigenvectors are linearly independent. (f) The sum of two eigenvalues of a linear operator T is also an eigenvalue of T. (g) Linear operators on infinite-dimensional vector spaces never have eigenvalues. (h) An n x n matrix A with entries from a field F is similar to a diagonal matrix if and only if there is a basis for Fn consisting of eigenvectors of A. (i) Similar matrices always have the same eigenvalues, (j) Similar matrices always have the same eigenvectors, (k) The sum of two eigenvectors of an operator T is always an eigenvector of T.