课程名称︰线性代数一 课程性质︰必修 课程教师︰陈其诚 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰99/12/24 考试时限（分钟）：100 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : Write your answer on the answer sheet. 1.(25 points) Solve the following system of linear equations. 2x+2y+v=0 x+y-w=0 z+3w=0 x+y-z-4w=0 x+y+z+v+4w=0 2.(25 points) Calculate det(A), where ┌ ┐ │1 2 3 4 5│ │2 3 4 5 1│ │3 4 5 1 2│ │4 5 1 2 3│ │5 1 2 3 4│ └ ┘ 3. Either give a brief reason or give a counter example for each of the following assertions (5 points each): (a) If (a1,a2,a3,a4), (b1,b2,b3,b4)∈R^4 are linearly independent, then 4 4 the solution set W of the system of equations Σaixi=0 and Σbixi=0 i=1 i=1 is a 2-dimensional subspace of R^4. (b) If W is a 2-dimensional subspace of R^4, then there exist linearly independent (a1,a2,a3,a4), (b1,b2,b3,b4)∈R^4 such that W is the 4 4 solution set of the system of equations Σaixi=0 and Σbixi=0 i=1 i=1 (c) An n-linear function f:M (F)→F must be linear. n*n (d) Suppose f,g:M (F)→F are alternating n-linear functions with f(I)=g(I), n*n then f(A)=g(A) for all A. (e) If two n*n matrices A and B have the same reduced row echelon form, then they must be similar. 4. Give a rigorous proof for (a) and answer (b) with rigorous reasoning. (a)(15 points) For any A∈M (F), det(A^t)=det(A). n*n (b)(10 points) Let e1,...,e5 be the standard basis of R^5. It is known that fi=ei+e4, i=1,2,3,5, form a basis of W = {(x,y,z,v,w)∈R^5 | x+y+z-v+w=0 } Denote a=(1,2,3,10,4)∈W. Find all subsets {b,c,d}⊂{f1,f2,f3,f5} such that the vectors a,b,c,d form a basis of W (give the complete list of all these subsets). --

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