課程名稱︰線性代數 課程性質︰電機系必修 課程教師︰馮世邁 開課學院：電資院 開課系所︰電機系 考試日期（年月日）︰2011.3.31 考試時限（分鐘）：50min 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1.Let the 3*5 matrix A and the vector b be respectively defined by [ 1 -1 -3 1 -1 ] [ -2 ] A=[a1 a2 a3 a4 a5]=[ -2 2 6 0 -6 ], b=[ -6 ] [ 3 -2 -8 3 -5 ] [ -7 ] (a)(15%)Find the reduced row echelon form of [A b]. (b)( 9%)Find the general solution to Ax=b in vector form. (c)( 6%)What are the rank and nullity of A? (d)(10%)Find all pivot columns of A. Write each column of A in terms of the pivot columns of A. (You may express your answer in terms of a1,a2,a3,a4 ,a5.) (e)( 5%)From your result in Part(a), find the reduced row echelon form of B , where B = [ a1 a2 a3 a4 a5 ]. (f)(10%)Using your result in Part(a), choose 3 column vectors form A to form a 3*3 matrix A' so that A'x = b has a unique solution. Find the solution to A'x = b. 2.Let T: R^3 -> R^3 be the linear transformation defined by [ x1 ] [ 2x2 +rx3 ] T( [ x2 ] ) = [ x1 - x2 +2x3 ] [ x3 ] [ 2x1 + tx2 +3x3 ] (a)( 5%)Find the standard matrix A of T. (b)( 5%)Find the values of r and t such that T is not onto. (c)( 5%)Can we find r and t such that rank A = 1? Explain your answer. 3.(20%)Let A be the standard matrix in Problem 2(a). Let r=-1 and t=-1. Find the inverse A^-1. 4.(10%)Let Q be an m*m invertible matrix. Show that the two m*n matrices, A and QA, have the same reduced row echelon form. --

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