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國立新竹教育大學九十八學年度研究所碩士班招生考試試題 所別：應用數學系碩士班 科目：線性代數（本科總分：150 分） ※ 請橫書作答 1 1 1 1 1 0 0 1 1. Let A1 , A2 , A3 and A4 . 1 0 0 1 1 1 1 1 (a) Prove that A1 , A2 , A3 and A4 form a basis for M 22 () . (10 分) a a (b) Find the unique representation of arbitrary vector 1 2 in a3 a4 M 22 () as a linear combination of A1 , A2 , A3 and A4 . (10 分) (c) What is the dimension of M 22 () ? (5 分) 2. Let c0 , c1 and c2 be three distinct scalars in an infinite field F . (a) Describe the Lagrange interpolation polynomial f associated with c0 , c1 and c2 . (10分) (b) Use the Lagrange interpolation formula to construct the polynomial of smallest degree whose graph contains the points (-4,24), (1,9) and (3,3). (10分) 3 3. Determine whether the following sets are subspaces of under the operations of addition and scalar multiplication defined on 3 . Justify your answers. (a) W1 (a1 , a2 , a3 ) 3 : a1 4a2 a3 0 . (10 分) (b) W2 (a1 , a2 , a3 ) 3 : 5a12 3a2 2 6a32 0 . (10 分) 5 1 0 4. Let A 1 5 0 . 0 0 4 (a) Find the eigenvalues of A . (5 分) (b) Find the algebraic and geometric multiplicity with respect to the eigenvalues for A . (5 分) (c) Find an orthogonal matrix P such that P 1 AP is diagonal. (10 分) 第 1 頁，共 2 頁 1 3 3 5. Let A 7 2 9 . 2 1 4 (a) Find the eigenvalues of A . (5 分) (b) Find the algebraic and geometric multiplicity with respect to the eigenvalues for A . (5 分) (c) Find the Jordan canonical form of A . (10 分) 6. Let T : 2 2 be the linear transformation defined by T ( x, y ) ( x 2 y , 2 x y ) Find an orthonormal basis of eigenvectors of T for 2 . (10 分) 7. Prove or disprove: If a matrix A is positive definite, then the eigenvalues of A is positive. (10 分) 8. Find a normalized QR decomposition for A and use it to solve the least-squares problem Ax y , where 1 2 2 A 2 1 and y 3 1 1 0 (15 分) 9. Let V be an inner product space and u, v V . Prove that u, v u v (the Cauchy-Schwarz inequality). (10 分) 第 2 頁，共 2 頁