课程名称︰线性代数 课程性质︰必修 课程教师︰冯蚁刚(不过其实四个老师考题都一样) 开课学院：电资学院 开课系所︰电机系一年级 考试日期（年月日）︰2009 5/21 考试时限（分钟）：50分钟 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 1.Let A be the matrix defined by 2 1 1 A= 1 2 1 (注:这是方阵) 1 1 2 (a)(10%)Find the characteristic polynomial of A. (b)(20%)Find an invertible matrix P and a diagonal matrix D such that A=PDP^-1 (c)(10%)Does there exist a real matrix B such that A=B^2? If it does, find one such B.Is B unique?(You can express your answer in terms of P,P^-1,and D.) 2.Define the following vectors: 1 2 2 4 u1= 0 , u2= 1 , u3= -1 , v= 0 . -1 -1 -1 2 1 0 3 1 (注:上面是四个向量) (a)(20%)Find an orthogonal basis B={w1,w2,w3}for Span{u1,u2,u3}. (b)(10%)The vector v is in Span {u1,u2,u3}.Find c1,c2,c3 such that v=c1w1+c2w2+c3w3 (c)(10%)Find a vector w4 such that {w1,w2,w3,w4}forms an orthogonal basis for R^4. 3.(10%)Let the characteristic polynomials of the matrices A, B and C be respectively fA(t)=-t(t^2-1), fB(t)=-(t+1)(t^2+1), fC(t)=-(t+2)(t^2-1) Which matrix is not invertible? Which matrix is not diagonalizable? Which matrix is both invertible and diagonalizable? Explain your answer. (Answers without explanation get 0%.) 4.(10%)Let B1={v1,v2}and B2={v3,v4} be respectively a basis for W1 and W2 (subspaces of R^n). Prove that if the intersection of W1 and W2 is the zero subspace (i.e.,W1∩W2={0}),then {v1,v2,v3,v4} is linearly independent. --

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