课程名称︰工程数学-线性代数
课程性质︰电机系必修
课程教师︰冯世迈
开课学院：电机资讯学院
开课系所︰电机系
考试日期（年月日）︰2015.04.09
考试时限（分钟）：50分钟
试题 :
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.