試題詳解

17. The linear independence is without doubt one of the most important concepts in linear algebra. Which of the following statements about this concept are true?
(A) The number of linearly independent columns of any matrix is equal to the number of linearly independent rows of the matrix.
(B) A set of vectors {v₁, ..., } is linearly independent if there exist coefficients c₁, ..., , all of them are zero, such that c₁v₁ + ... + = 0.
(C) The linear independence of a set of vectors is a necessary but not sufficient condition for them to be a basis of a vector space.
(D) A square matrix is diagonalizable if and only if all its eigenvectors are linearly independent.
(E) Let (λ₁, x₁) be the i th eigenvalue-eigenvector-pair of a square matrix. Then all λ's are distinct if and only if the set {x₁, ..., } of eigenvectors is linearly independent.