课程名称︰线性代数 课程性质︰ 课程教师︰冯蚁刚 开课学院：电资学院 开课系所︰电机系 考试日期（年月日）︰2008/03/20 考试时限（分钟）：50 是否需发放奖励金：是，谢谢 （如未明确表示，则不予发放） 试题 : 1. Determine whether the given vector v is in the span of [ -1 ] [ 1 ] [ 1 ] S = { [ 5 ] ,[ 3 ] ,[ -1 ] } [ 2 ] ,[ 4 ] ,[ 1 ] T (a.) v = [ 5 3 11 ] (20 %) T (b.) v = [ 1 1 2 ] (10 %) If so, write v as linear combination of vectors in S. [ 1 -3 0 5 3 ] 2. Suppose that the reduced row echelon form of A is R = [ 0 0 1 2 -2 ] [ 0 0 0 0 0 ] , determine A if the first and third colums of A are [ 1 ] [ 2 ] a1 = [ -1 ] and a3 = [ 0 ], respectively. (30 %) [ 2 ] [ -1 ] 3. Let A and B be n * n matrices. We say that A is similar to B if B = P^(-1) A P for some invertible matrix P. (a.) Prove that if A is similar to B and B is similat to C, then A is similar to C. (b.) Suppose that A is similar to B. Prove that if A is invertible, then B is invertible, and A^(-1) is similar to B^(-1). (10%) 4. (a.) Prove that if A is a m * n matrix and B is an n * p matrix, then rank(AB) ≦ rank (B). ( Hint : Prove that if the k-th column of B is not a pivot column of B, then the k-th column of AB ois not a pivot column of AB.) (10 %) (b.) Prove that is A is a m * n matrix and P is an invertible m * m matrix, then rank(PA) = rank(A). ( Hint: Apply 4(a.) to PA and to P^(-1)(P A) ). (10%) --

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