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課程名稱︰應用數學一
課程性質︰系必修
課程教師︰陳義裕
開課學院：理學院
開課系所︰物理系
考試日期（年月日）︰98/06/09
考試時限（分鐘）：180分
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Calculators , dictionaries ( hardcopy or electronic ) , PDA , cell phones , etc
are NOT allowed during the exam .
1.(15 points)
Please compute the determinant of
┌ 1 2 1 1 -2 ┐
│ 3 -1 2 1 2 │
│ 3 1 -1 2 1 │
│ 4 0 1 2 2 │
└ 1 3 0 1 1 ┘
The answer should be an integer with a magnitude less than 20 . Please note
that you won't get any partial credits if you do not obtain the correct answer
2.(25 points)
A particle can be in one of two states : α or β . If it is initially in state
α , then at the next moment it has 1/4 probability to remain in state α and
3/4 probability to switch to state β . But if it is initially in state β ,
then at the next moment it has equal probability (=1/2) to be in state α or
β . Let pα(n) and pβ(n) denote the probability of the particle in state α
and β at the n-th moment , respectively . Define
→ ╭ pα(n) ╮ ^ ╭ 1/4 1/2 ╮
rn ≡ │ │and A ≡ │ │
╰ pβ(n) ╯ ╰ 3/4 1/2 ╯
→ ^ →
Then we clearly have r(n+1) = A(rn)
(a)(10 points) Please find all the eigenvalues and their associated
^
eigenvectors of the matrix A .
(b)(10 points) Please explain why we will end up having the same probabilistic
distribution when the "time" n goes to infinity , irrespective of whatever
initial distribution is given .
(c)(5 points) What is this final "equilibrium" probabilistic distribution ?
3.(35 points)
Let U be an inner product space of dimension n . The inner product is defined
over the complex numbers .
(a)(10 points) Please prove the "Cauchy-Schwarz inequality" , which asserts
that → → → → , and the equality holds only when → and →
│＜u│v＞│ ≦ ∥u∥∥v∥ u v
are linearly dependent .
(b)(5 points) Please prove "Gram-Schmidt process : We can always find an
orthonormal basis for U ."
(c)(15 points) The adjoint operator ^+ is defined via ^+→ → → ^ →
L ＜L(v)│u＞≡＜v│L(u)＞
→ → ^
for all vectors u and v . An operator L : U → U is said to be self-adjoint if
^+ ^
it satisfies L = L . Please prove that we can always find an orthonormal basis
of U to be the eigenvectors of a self-adjoint operator .
(d)(5 points) The following list is the matrix representation of 5 linear
operators in some orthonormal basis . Please identify ( no need to explain )
all the self-adjoint operator(s) , if any .
╭ 1 2i 4 ╮ ╭ √3 2 1 ╮ ╭ 1 2 4 ╮
A1 ≡ │ 2i -2 5 │ , A2 ≡ │ 2 -2i -4 │ , A3 ≡ │ 7 -2 7 │
╰ 4 5 3 ╯ ╰ 1 -4 3 ╯ ╰ 4 2 3 ╯
╭ 2 i 7 ╮ ╭ 2 1 -3i ╮
A4 ≡ │ 5 3 -i │ , A5 ≡ │ 1 √3 5 │ .
╰ 1 5 2 ╯ ╰ 3i 5 1 ╯
4.(25 points)
Let V be an inner product space of dimension n . The inner product is defined
over the real numbers .
^
(a)(10 points) A unitary operator F by definition satisfies
^ → ^ → → →
＜F(u)│F(u)＞=＜u│u＞
→ ^+^ ^
for all u in V . Please prove that a unitary operator satisfies F F = I , where
^
I is the identity operator in V .
(b)(5 points) The following is the matrix representation of two linear
operators in some orthonormal basis . Please identify (no need to explain) all
the unitary operator(s), if any .
╭ 1/√2 1/√2 ╮ ╭ 1/2 -√3/2 ╮
F1 ≡ │ │ , F2 ≡ │ │ .
╰ 1/√2 -1/√2 ╯ ╰ √3/2 1/2 ╯
^ 2 2
(c)(10 points) Can we have a unitary operator F : R → R such that in some as
yet unspecified basis it has the following matrix representation ╭ 1 1 ╮ ?
│ │
╰ 2 3 ╯
Please note that no partial credits will be granted unless you give the correct
answer and a complete explaination to it .
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