課程名稱︰應用數學一
課程性質︰物理系必修
課程教師︰李慶德
開課學院：理學院
開課系所︰物理系
考試日期（年月日）︰2012/06/18
考試時限（分鐘）：~210 min
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Do 6 and only 6 of the problems given below!
1. Let V be the subspace of C[a,b] spanned by 1, e^x, e^-x, and let D be the
differentiation operator on V.
(a) Find the transtition matrix S representing the change of coordinates
from the ordered basis [1, e^x, e^-x] to the ordered basis
[1, cosh(x), sinh(x)].
(b) Find the matrix A representing D with respect to the ordered basis
[1, cosh(x), sinh(x)].
(c) Find the matrix B representing D with respect to the ordered basis
[1, e^x, e^-x].
(d) How are the matrices A and B related?
2. Given a collection of points (x_1, y_1), (x_2, y_2), ..., (x_n, y_n), let
~ T ~ T
x = (x_1, x_2, ..., x_n) , y = (y_1, y_2, ..., y_n)
1 n 1 n
<x> = - Σ x_i , <y> = - Σ y_i.
n i=1 n i=1
And let y = c_0 _ c_1 x be the linear function that gives the best least
squares fit to the points. Assuming that <x> = 0, find the coefficients
c_0 and c_1 in terms of the quantities (i.e., vectors and averages)
defined above.
3. Let
- -
1 | 1 -1 |
A = - | 1 -1 |
2 | 1 1 |
| 1 1 |
- -
(a) Determine the projection matrix P that projects vectors in R^4 onto R(A).
(b) Determine the projection matrix Q that projects vectors in R^2 onto
R(A^T).
(c) What do the projection matrices I - P and I - Q do to vectors in
R^4 or R^2 ?
4. Let
- -
| 1 -2 -1 |
A = | 2 0 1 |
| 2 -4 2 |
| 4 0 0 |
- -
(a) Find an orthonormal basis for the column space of A.
(b) Compute the Gram-Schmidt QR factorisation of the matrix A.
5. Let
- - - - - -
~ | 1 | ~ | 1 | ~ | 0 |
b_1 = | 1 | b_2 = | 0 | b_3 = | 1 |
| 0 | | 1 | | 1 |
- - - - - -
and let L be the linear transformation from R^2 to R^3 defined by
~ ~ ~ ~
L( x ) = x_1 b_1 + x_2 b_2 + (x_1 + x_2) b_3
~ - -
for all x = | x_1 | in R^2.
| x_2 |
- -
Find the matrix A that represents L with respect to the ordered bases
~ ~ ~ ~ ~
{u_1, u_2} and {b_1, b_2, b_3}, where
~ T ~ T
u_1 = [1, 1], u_2 = [2, -1].
6. Consider the following Pauli matrices: (i = √-1)
- - - - - -
σ_x = | 0 1 | σ_y = | 0 -i | σ_z = | 1 0 |
| 1 0 | | i 0 | | 0 -1 |.
- - - - - -
With I denoting the 2×2 identity matrix and k, l = x, y, z, that satisfy
the relations σ_k σ_l + σ_l σ_k = 2δ_kl I, and σ_x σ_y = i σ_z
in which x, y, and z can be changed cyclically. Find e^(i φ σ_z),
e^(i φ σ_x), and e^(i φ σ_y) in the form of I f(φ) + σ_k g(φ),
for instance, or equivalently in the form of
- -
| f_1(φ) f_2(φ) |
| f_3(φ) f_4(φ) |
- -
7. Let
- -
| 2 0 1 0 |
A = | 0 2 0 1 |
| 1 0 2 0 |
| 0 1 0 2 | .
- -
(a) Compute the eigenvalues and eigenvectors of A.
(b) Find an unitary (orthogonal) matrix that diagonises A.
(c) Comment on why A is diagonalisable by a unitary (or orthogonal) matrix.
8. Three masses m_1, m_2, and m_3 moving on a frictionless plane are connected
by four springs with spring constant k. The equations that describe the
motion of this system are given by (with primes denoting derivatives with
respect to time t)
m_1 x_1'' = -2 k x_1 + k x_2
m_2 x_2'' = k x_1 - 2 k x_2 + k x_3
m_3 x_3'' = k x_2 - 2 k x_3
Solve the system if m_1 = m_3 = 1/3, m_2 = 1/4, k = 1, and
x_1 (0) = x_2 (0) = x_3 (0) = 1
x_1' (0) = x_2' (0) = x_3' (0) = 0
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◆ From: 140.112.4.202
※ 編輯: Chao33 來自: 140.112.4.202 (07/01 23:30)
※ 編輯: Chao33 來自: 140.112.4.202 (07/01 23:33)