試題詳解

8、Let V together with the inner product (•, •) v be an inner product space over R. For a
collection of N linearly independent vectors {w1, ..., w_N} in V, let M = [m_i，,j] ∈R^{N✖ N}be
the corresponding Gram matrix, i.e., the (i, j)-th entry of M is given by m{i_，,j} = (wi, wj)V.
Define the following set of real-valued functions f: V→R

which is a vector space over R when combined with standard addition and scalar multiplica-
tion of real-valued functions. For any
in the vector space F, define

Which of the following statements is/are true?
(A) M is positive definite.
(B) If m_1,1 = 2, m_2,2 = 1, and m_1,2 = -, then ||w₁ - w₂||V = 4, where ||\cdot||_V is the vector norm induced by the inner product(•, •) v.
(C) (f, g)_F is an inner product for elements f, g∈ F.
(D) Given g(y) = (v1, y)V, we have (f, g)_F = f(w1) for all f∈ F.

(E) None of the above is true.