課程名稱︰工程數學-線性代數 課程性質︰電機系必修 課程教師︰馮世邁 開課學院：電機資訊學院 開課系所︰電機系 考試日期（年月日）︰2015.04.09 考試時限（分鐘）：50分鐘 試題 : 1. Let A and b be respectively defined by [ 1 -1 1 0 1 ] [ -4] A= [ 1 -1 2 2 0 ] ,b=[ -5] [ 3 -3 2 -2 0 ] [-11] (a)(15%)Find the reduced row echelon form of [A b]. (b)( 6%)What are the rank and nullity of A? (c)( 9%)Find the general solution to Ax=b in vector form. (d)( 6%)Write each non pivot column of A as a linear combination of the pivot columns of A. (e)( 4%)Without using the Gaussian elimination, find the general solution to Ax=b+a1. (f)(5%*4)Let T:R^5 -> R^3 be defined by T(x)=Ax. (i) Is T onto? (ii) Is T ont-to-one? (iii)Find the range of T. (iv) Find the null space of T. 2.(15%)Find the inverse of the following matrix: [ 1 -1 1] [ 2 -1 4] [ 2 -2 3] 3. Let R be the reduced row echelon form of a 3x4 matrix A and let rank A = 3. Find the reduced row echelon form of the following matrices (you may express your answer in terms of I,O,R,A): (NO EXPLAINATION IS NEEDED.) (a)(5%) [2A A] ;(b)(5%) [A ] ;(c)(5%) R(Transpose). [I4] 4.(10%)Prove that two mxn matrices A and B have the same reduced row echelon form if and only if there exists an mxm invertible Q such that B=QA. --

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