海洋大学河海工程学系2003工程数学(二)2B班期中考.pdf

班級:ゼ飽
海洋大學河海工程學系工程數學 Z

1. Given the three matrices (20%)
 1 3  1 3  1 3 
−−
Q =  2 2 , Q =  2 2 , Q =  2 2 
1 3 1 2 3 1 3 3 1
−−−
 2 2  2 2  2 2 
and a position vector y , write down the geometric relationship of
~
y
Q2 y Q1 y
(a) Q1 y and y . ~ ~
~ ~
Q3 y
~
(b) Q2 y and y . y
~ ~ ~
(c) Q3 y and y . x
~ ~
(d) Can you find a matrix Q such that QT Q = I and Q = QT ?
T
(e) Q1Q1 = ?
T
(f) Q2 Q2 = ?
(g) Q3Q3 = ?
(h) det Q1
(i) det Q2
(j) det Q3

2. Given A x = b ,
~ ~
1 3 4  8 

where A = 2 1 5  and b =  8 .

7 6 1932
(a) Find the rank of []A . (5%)
(b) Find the nontrivial solution {φ} such that
[]A T φ= 0 ,
~
where T denotes the transpose. (5%)
(c) Determine {b}{}T φ= ? (5%)
(d) Solve the general solution of x (10%)
~
(e) Please write down the Fredholm Alternative Theorem using this example. (5%)
3. (a) Given the equation x 2 − xy + y 2 = 1, is the shape ellipse, hyperbolic or
parabolic curve ? (5%)
2 2 a11 a12 x
(b) Transform x − xy + y = 1 to quadratic form {}x y = 1.
a21 a22 y
a11 a12 
Find the symmetric matrice A = (5%)
a21 a22 
(c) Find the eigenvalues (λ1 , λ2 ) and eigenvectors (v1 , v2 ) of A. (10%)
2 2 2 2 xx
(d) Transform x − xy + y = 1 to λ1