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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