試題詳解

5、Let V be the vector space of all A ∈ R^m✖R, with the operations of matrix addition and multiplication of a matrix by a real scalar. Let T be a linear transformation from V to V, and {B₁, B₂,..., B_n} be a basis for V. Suppose U is a linear transformation from V to Rn given by U(c₁B₁ + c₂B₂ + ... + c_nB_n) =. Which of the following statements is/are true?
(A) Let {A₁, A₂,..., A_k} be linearly independent over R. The set {U(A₁), U(A₂),…, U(A_k)} can be linearly dependent over R.
(B) Suppose the dimension of the range space of T is k over R. Then {T(B₁), T(B₂),……, T(B_k)} are linearly independent over R.
(C) n = m.
(D) Let c = U(B₁) and a = U(T(B₁), then aTa = cTc.
(E) None of the above is true.