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）︰2010/11/19 考试时限（分钟）：120分钟 是否需发放奖励金：是 (1) (25 points) Calculate A^(-1), where ┌ ┐ │2 0 3 -2 │ A = │1 1 1 0 │ │1 1 1 1 │ │1 0 2 0 │ └ ┘ (2) (25 points) Let β={v1,v2,v3,v4} be the ordered basis of R^4, with v1=(2,1,1,1), v2=(0,1,1,0), v3=(3,1,1,2), v4=(-2,0,1,0). Let T:R^4→R^4 be the linear transformation sending (x1,x2,x3,x4)∈R^4. to (2x1+x3,x1+x2+x3+9x4,3x2+3x4,x1-x2+x3-x4)∈R^4. Calculate the matrix [T]_β (3) Either give a brief reason or give a counter example for each of the following assertions (5 points each): (a) If A is a 5*4 matrix and B is a 4*5 matrix, then the 5*5 matrix AB can not be invertible. (b) If A is a 5*4 matrix and B is a 4*5 matrix, then the 4*4 matrix BA can not be invertible. (c) If T,U:V→W are both linear and agree on a basis for V, then T=U. (d) Every change of coordinate matrix is invertible. (e) An n*n matrix having rank n is invertible. (4) Give a rigorous proof for each of the following assertions. (a) (15 points) Let V and W be finite-dimensional vector spaces over a field F with ordered bases β and γ, respectively. For any linear transformation T:V→W, the mapping T^t:W*→V* defined by T^t(g)=gT for all g∈W* is a linear transformation with the property that β* γ [T^t] = ( [T] )^t γ* β (b) (10 points) Suppose A is a 5*5 matrix satisfying A^5=0 and A^4≠0. (i) (5 points) Then A has rank 4 (Hint: There is some x∈F^5 such that {x,Ax,(A^2)x,(A^3)x,(A^4)x}⊂F^5 is linearly independent. ) (ii) Then I-A is invertible (Hint: The kernal of L_(I-a) is trivial). --

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