複選題9、Which of the following statements is/are true?(A) Let x and y be two non-zero vectors in Rᵀ. The matrix xyᵀ is diagonalizable ifyᵀx ≠ 0.(B) If A ∈ Rⁿˣⁿ is symmetric and positive definite, then A + A⁻¹ - 2Iₙ is positive semi- definite.(C) For u, v ∈ Rⁿ such that uᵀv ≠ 0, Rⁿ = (span{u})⊥⊕span{u}, where ⊕ is theoperation of direct sum of two vector spaces.(D) For the real matrix Q = Iₙ - uuᵀ, where u ∈ Rⁿ, we have det(Q) + rank(Q) = n.(E) None of the above is true.

複選題

9、Which of the following statements is/are true?
(A) Let x and y be two non-zero vectors in Rᵀ. The matrix xyᵀ is diagonalizable if
yᵀx ≠ 0.
(B) If A ∈ Rⁿˣⁿ is symmetric and positive definite, then A + A⁻¹ - 2Iₙ is positive semi- definite.
(C) For u, v ∈ Rⁿ such that uᵀv ≠ 0, Rⁿ = (span{u})^⊥⊕span{u}, where ⊕ is the
operation of direct sum of two vector spaces.
(D) For the real matrix Q = Iₙ - uuᵀ, where u ∈ Rⁿ, we have det(Q) + rank(Q) = n.

(E) None of the above is true.

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複選題8、Let V together with the inner product (•, •) v be an inner product space over R. For acollection of N linearly independent vectors {w1, ..., wN} in V, let M = [mi，,j] ∈RN✖ Nbethe 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→Rwhich is a vector space over R when combined with standard addition and scalar multiplica-tion of real-valued functions. For anyin the vector space F, defineWhich of the following statements is/are true?(A) M is positive definite.(B) If m1,1 = 2, m2,2 = 1, and m1,2 = -, 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.

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