[試題] 100下 黃維信 線性代數 期中考
課程名稱︰線性代數
課程性質︰選修，資訊計算組必修
課程教師︰黃維信
開課學院：工學院
開課系所︰工科海洋
考試日期（年月日）︰2012/4/18
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 Examniation
1.Find the symmetric factorization A=LDL^T of (15%)
A=┌1 2 0 ┐
2 6 4
0 4 11
└ ┘
2.Suppose R is rectangular (m by n)and A is symmetric(m by m).(15%)
(a)Show R^TAR is symmetric What shape is this matrix?
(b)Show R^TR has no negative numbers on its diagonal.
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,x3..)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.Write all known relations between r and m and n if AX=b has(15%)
(a)no solution for some b.
(b)infinitely many solutions for every b.
(c)exactly one solution for some b, no sloution for other b.
(d)exactly one solution for every b.
5.Suppose all vectors x in the unit square 0≦X1≦1, 0≦X2≦1 are transformed
to Ax(A is 2 by 2).(15%)
(a) What is the shape of the transformed region?
(b) For which matrices A is that region a square?
(c) For which A is it a line?
(d) For which A is the new area still 1?
6.Use Gram-schmidt to construct an orthonormal pair q1,q2 form a1=(4,5,2,2)
and a2=(1,2,0,0). Express a1 and a2 as combination of q1 and q2, and find the
triangular R in A=QR.(10%)
7.Find a straight line which gives the best fit to the curve y=x^3 between x=0
and x=1 by using orthogonality.(15%)
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