课程名称︰工程数学-线性代数 课程性质︰必修 课程教师︰冯蚁刚 开课学院：电机学院 开课系所︰电机系 考试日期（年月日）︰2011.03.24 考试时限（分钟）：55 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : T(A)表示为A的转置矩阵 A(ij)表示A中的第i row 第j column。 1.Judge if the following statements are true or false. Give a concise proof to each true statement, and a counterexample to each false statement. (a) If X and Y are invertible n*n matrices, then X+Y is also invertible.(20%) (b) Every linear combination of the rows of a matrix B is also a linear combination of the rows of the reduced row echelon form of B. (20%) 2. Suppose R is the reduced row echelon form of an m*n matrix A with rank r, and the reduced echelon form of T(R) is the n*m matrix Q=q(ij). Show that q(ii)=1 for i = 1,...,r, and q(ij)=0 elsewhere. (25%) Use the above result to deduce first that rank T(A) 小於等於 rank A. Then deduce further that rank T(A)= rank A. ( 5%) (Hint: Note there exist an invertible matrix P and elementary matrices E1,...,Ek such that A = PR and Ek...E1T(R)T(P)=QT(P) ) 3. For 1 0 -1 -1 0 -1 1 -9 S={[-1],[ 2],[ 1],[ c]},find value(s) of c making S linearly dependent.(30%) 1 1 0 2 -- 没有伴随着痛苦的教训是没意义的。 人如果不牺牲一些东西，就无法得到任何东西。 但是当超越了障碍，并且把得到的东西变成属於自己的东西时... 人应该就能够得到无法取代的钢之心灵吧。 <Fullmetal Alchemist> --

※ 发信站: 批踢踢实业坊(ptt.cc) ◆ From: 140.112.4.186