課程名稱︰線性代數
課程性質︰資工系大二系必修
課程教師︰陳文進
開課學院：電資學院
開課系所︰資工系
考試時間︰2006.11.21 14:20-17:00
是否需發放獎勵金：是，謝謝 ^^*
（如未明確表示，則不予發放）
試題 :
線性代數期中考
(1) (22%)Let A be a square n*n matrix, which of the following statements are
equivalent to the statement that A is non-singular?
A. A^T is an invertible matrix.
B. A is an invertible matrix.
C. A is row equivalent to the n*n identity matrix.
D. There is an n*n matrix C such that CA=I
E. There is an n*n matrix D such that AD=I
F. A has n pivot positions.
G. The equation Ax=b has exact one solution for each b in R^n.
H. The columns of A form a linearly independent set.
I. The columns of A span R^n.
J. The linear transformation x→Ax is one-to-one.
K. The linear transformation x→Ax maps R^n onto R^n.
L. The equation Ax=0 has only the trivial solution.
M. The columns of A form a basis of R^n.
N. rank A=n.
O. dim C(A)=n.
P. C(A)=R^n.
Q. N(A)={0}.
R. dim N(A)=0.
S. The equation Ax=b has at least one solution for each b in R^n.
T. C(A)┴ ={0}.
U. N(A)┴ =R^n.
V. R(A) =R^n.
(2) (15%)Find the values of a, b such that the following linear equations
are inconsistent.
{ x1 + x2 - 2x3 + 3x4 = 0
{3x1 + 2x2 + ax3 + 7x4 = 1
{ x1 - x2 - 6x3 - x4 = 2b
(3) (15%)Find the distance from the origin to the hyper-plane in R^4 spanned
by (1,-1,1,-1), (1,1,-1,-1), (1,-1,-1,1)and passing through (2,1,0,1).
(4) (8%)If A and B are 3*3 matrices satisfying AB = 2A + B and
[1 0 0]
I = [0 1 0] is the 3*3 identity matrix. Find (A-I)^-1.
[0 0 1]
(5) (20%)If {a1,a2,a3) is a basis for vector space V, which of the following
are basis for V also?
(i)
A. {a1+a2, a2+a3, a3+a1}
B. {a1+a2, a2+a3, a3-a1}
C. {a1+a2, a2+2a3, a1+2a2+2a3}
D. {a1+2a2, 2a2-3a3, a1+3a3}
E. {a1+a2+a3, a1-a2+a3, a1-2a3}
(ii)
For each of the basis found in (i), find the coordinates of the vector
x = a1+2a2+a3 with respect to that basis.
(6) (60%)Find the bases for R(A), C(A), N(A), and N(A┴), where
[ 1 1 0 1 -1]
[ 1 1 2 -1 1]
A =[ 2 2 2 0 0]
[-1 -1 2 -3 3]
試題結束
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.112.107.116