Label the following statements as true or false. (a) Every linear operator on an n-dimensional vector space has n dis-tinct eigenV'alues. (b) If a real matrix has one eigenvector, then it has an infinite numberof 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 eigen-value ofT. (g) Linear operators on infinite-dimensional vector spaces never haveeigenvalues.(h ) An n x n matrix A with entries from a field F is similar to adiagonal matrix if and only if there is a basis for Fn consisting ofeigenvectors 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 eigen-vector ofT.
Label the following statements as true or false.
(a) Every linear operator on an n-dimensional
tinct eigenV'alues.
(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 eigen-
value ofT.
(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 eigen-
vector ofT.
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