课程名称︰线性代数 课程性质︰选修 课程教师︰黄维信 开课学院：工学院 开课系所︰工程科学与海洋工程学系 考试日期（年月日）︰2008/4/25 考试时限（分钟）：120 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 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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