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題組內容

Let V be the set of all 2 x 2 symmetric matrices and B =.Let T be a linear operator on V delined by T(A) = B^TAB, for each A V.

(b) (12%) Find the eigenvalues of T.

其他申論題

(3)立法院院會因高通案而邀請公平會主委丙到場備詢; 請從憲法規定與學理分別討論甲、乙、丙是否有義務前往備詢?又是否有義務回答所有問題?(60分)

(a) (4%) The matrix representation of T' relative to ordered bases B, C.

(b) (4%) The matrix representation of T relative to ordered bases B, D.

(a) (3%) Find a basis for V.

(a) If A and B are n x n matrices that have the same rank, then the rank of A? must equal the rank of B2.

(b) Let L : R2→ R2 be a linear operator, and let A be the standard matrix representation of L. If L2 is defned by L2(x) = L(L(x)) for all x R2 then L2 is a linear operator and its standard matrix representation is A2.

(c) If L1 and L2 are both linear operators on a vector space V, then Li + L2 is also a linear operator on V, where L1 + L2 is the mapping defined by (L1+L2)(v)=L1(v)+L2(v)forallvV

(d) If N(A) = {0}, then the system Ax = b will have a unique least squares solution.

(e) If{u1, u2,... ,uk} is an orthonormal set of vectors inRnand U = {u1, u2,... ,uk} then UUT = In (the n x n identity matrix).

(15%) Let T:P2(R)＞ P2(R) be a linear operator that is defined according to T(g(x))=g(x)+ , in which P2(R) is a set of all polynomials with real-value coefficients and degree n, n=0,1,2. Please find the eigenvalues and the associated eigenvectors of the operator

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