→ james11022:好難喔，果然是台大的實力 05/10 14:59
課程名稱︰線性代數
課程性質︰選修
課程教師︰黃維信
開課學院：工學院
開課系所︰工程科學與海洋工程學系
考試日期（年月日）︰2008/4/25
考試時限（分鐘）：120
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Linear Algebra
Midterm Examination
1.find the pivots and the solution for these four equations: (10%)
2x + y = 0
x + 2y + z = 0
y + 2z + t = 0
z + 2t = 5.
2.true or false, with reason if true and counterexample if false:(25%)
(a) if L1U1=L2U2 (upper triangular U's with nonzero diagonal, lower
triangular L's with unit diagonal), then L1=L2 and U1=U2.
the LU factorization is unique.
(b) if A^2 + A = I then A^-1 = A + I.
(c) if all diagonal entries of A are zero, then A is singular.
(d) if V is orthogonal to W, then V┴ is orthogonal to W┴.
(e) V orthogonal to W and W orthogonal to Z makes V orthogonal to Z.
3.which of the following are subspaces of R∞? (15%)
(a) all sequences like (1,0,1,0,...) that include infinitely many zeros.
(b) all sequences (x1,x2,...) with xj=0 from some point onward.
(c) all decreasing sequences : xj+1≦xj for each j.
(d) all convergent sequences : the xj have a limit as j→∞.
(e) all geometric progressions (x1,kx1,k^2x1,...) allowing all k and
x1.
4.under what conditions on b, Ax=b has a solution for the following A and
b?
┌1 2 0 3┐ ┌b1┐
A = │0 0 0 0│and b = │b2│. find a basis for the nullspace of A.
└2 4 0 1┘ └b3┘
find the general solution to Ax=b, when a solution exists. find a basis
for the column space of A. (20%)
5.find orthogonal vectors A,B,C by Gram-Schmidt form a,b,c:a=(1,-1,0,0),
b=(0,1,-1,0), c=(0,0,1,-1). (10%)
6.find a best approximation to y = x^5 by a straight line between x=-1
and x=1. (20%)
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