課程名稱︰線性代數 課程性質︰工管系科組大二必修 課程教師︰蘇柏青 開課學院：管理學院 開課系所︰工管系科組 考試日期（年月日）︰2010/11/12 考試時限（分鐘）：130 mins (2:40~4:50) 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1.(40%)(Gaussian Elimination and Elementary Matrices) Consider the following system of linear equations y-2z = 0 x+3y+2z = 2 -x-4y+2z = 2 (a) (1%) Write down the coefficient matrix A. (b) (1%) Write down the augmented matrix [ A∣b ]. (c) (2%) Do the elementary row operation R1 ←→ R2 on the augmented matrix to obtain [ A1∣b1 ]. (d) (2%) Find an elementary matrix E1 such that E1 [ A∣b ] = [ A1∣b1 ]. (e) (6%) Perform elementary row operations R1,3(k2) , R2,3(k3) , R3(k4) on [ A1∣b1] and obtain [ A2∣b2 ],[ A3∣b3 ], and [ A4∣b4 ] , respectively, so that [ A4∣b4 ] is in row-echelon form. Write down these three augmented matrices and find constants k2,k3,and k4. (f) (6%) Find elementary matrices E2,E3 and E4 so that Ei[ A(i-1)∣b(i-1) ] = [ Ai∣bi ] ,i=2,3,4. (g) (6%) Perform elementary row operations R2,1(k5) , R3,1(k6) , R3,2(k7) on [ A4∣b4 ] and obtain [ A5∣b5 ],[ A6∣b6 ], and [ A7∣b7 ], respectively, such that [ A7∣b7 ] is in reduced row-echelon form. Write down these three augmented matrices and find constants k5,k6, and k7. (h) (2%) Find the solution set of this system of linear equations. (i) (6%) Find elementary matrices E5,E6 ,and E7 such that Ei[ A(i-1)∣b(i-1)] = [ Ai∣bi ],i=5,6,7. (j) (4%) Find A^-1. (Hint: Use the fact that it is the product of elemetary matrices.) 2.(30%)(Matrix Operation and Determinants) Let A = ┌ 3 7 ┐, B = ┌ 6 -1 -2 ┐ └ 2 5 ┘ └ -1 3 1 ┘ ┌ 2 1 ┐,D =┌ 2 -1 ┐ C =│ 3 4 │ └ -5 2 ┘ └ -1 1 ┘ (a)(6%) Calculate A + D, B + C^T, BC, CB, AD, and DA. (b)(2%) Calculate B(B^T) and (B^T)B. (c)(4%) Find A^-1 and D^-1. (d)(3%) Calculate det(BC) and det(CB). (e)(2%) Calculate det(A) and det(D). (f)(1%) Calculate det(AD). (g)(1%) Calculate∣5 2∣. ∣1 2∣ ∣ 2 -4 3∣ ∣ 20 -40 30∣ (h)(4%) Calculate∣ 1 2 4∣ and ∣ 10 20 40∣. ∣-2 1 2∣ ∣-20 10 20∣ ∣5 4 3 2∣ (i)(3%) Calculate∣0 2 2 3∣. ∣0 0 -1 1∣ ∣0 0 0 7∣ ∣1 2 8 -1 -2∣ ∣0 1 2 3 2∣ (j)(4%) Calculate∣0 0 2 6 3∣ ∣0 0 0 4 1∣ ∣0 0 0 0 1∣ 3.(30%)(Vector Spaces) A set V is called a vector space over the set of real number R with two operations, namely vector addition "+" and scalar multiplication"·", if and only if the following 10 axioms are satisfied. For any elements u,v,w ε V and scalars c,d ε R, (1) u + v ε V. (2) u + v = v + u. (3) (u + v) + w = u + (v + w). (4) There exists a zero vector 0 ε V such that for any u ε V, u + 0 = u. (5) For any u ε V there exists a vector x ε V such that u + x = 0. (6) c·u ε V. (7) c·(u + v) = c·u + c·v. (8) (c + d) ·u = c·u + d·u. (9) (cd)·u = c·(d·u). (10) 1·u = u. (a)(10%) Let V = R^2. For any elements u = (u1,u2), v = (v1,v2) ε V and any scalar c ε R, define operations + and ·as follows. u + v = (u1 + v1,2*u2 + 2*v2), and cu = (c*u1,c*u2). Does (V,R,+,·) constitute a vector space? If so, verify that all 10 axioms are satisfied. Otherwise, give a counterexample that violates one of these axioms. (b)(10%) Let (V,R,+,·) be a vector space. Use the ten axioms only to prove that for any scalar c ε R, c0 = 0, where 0 is the zero vector in V. (c)(10%) Let V = R^2 and consider the vector space (V,R,+,·). Determine whether each of the following subsets of V is linear independent. (1) S = {(0,1)}. (2) S = {(0,1),(1,0)}. (3) S = {(1,2),(3,4)}. (4) S = {(1,2),(0,0)}. (5) S = {(1,2),(3,4),(5,6)} --

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