推 kuoly1 :也太快= = 03/31 19:28
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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