课程名称︰线性代数 课程性质︰系必修 课程教师︰冯世迈 开课学院：电资学院 开课系所︰电机系 考试日期（年月日）︰99年3月25日 考试时限（分钟）：50 min 是否需发放奖励金：是!感谢! 试题 : 1.Let the 3x5 matrix A and the vector b be respectively defined by [ 1 0 -3 -1 -2] [1] A = [a1 a2 a3 a4 a5]= [ 2 -1 -8 -1 -5] , b = [0] [-1 1 5 1 4] [2] (a)(20%) Find a lower triangular L and an upper triangular matrix U such that [A b] = LU. (b)(10%) Find the reduced row echelin form [R c] of [A b]. (c)( 5%) What are the rank and nullity of A? (d)(10%) Find the general solution to Ax = b in vector form. (e)( 5%) Find the general solution to Ax = 2b in vector form. (f)( 5%) Choose three column vectors from A to form a 3x3 matrix A' so that A'x = b is inconsistent. (g)( 5%) Let S = {a1, a2, a3, a4, a5}. Find a linearly independent subset S' of S such that Span S' = Span S. 2.(20%) Find the inverse of B = [a1 a2 a4] where a(i) are the column vectors defined in Porblem 1. 3.(10%) Let S = {u1, u2, ..., u(k)} be a linearly independent subset of R^n. Show that every vector in the span of S can be uniquely written as a linear combination of the vectors in S. 4.(10%) Let A be a 3x4 matrix with reduced row echelon form R. And the rand of A is 2. Find the reduced row echelon form of (a) R(Transpose) (b) the 6x4 matrix [A] [A] (You may express your answer in terms of A, R, O, and I.) --

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