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%) -- *╮ ╭ ╰─┼──┼─╯ \|/ （(⊙)） 我們是植栽,園藝 ╰─┼╮╰┼─╮ ╰┼╮╰┼─╮ ╰──╮╰∪╯ 不是插花喔 ╭─┬╯ ╮ ◎ ╰┼╮│ ╱ ╰─┼─╮ │ │ ＊ ╰┼╯＊╯ │ ╱§╲ │ 社版 Ntuflower │ ├～╯ ╰┴╯──╯╰ ∮ │ ╯ ╰──╯ ╰────★ ╯ ╰─┴─╯ --

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