課程名稱︰應用代數
課程性質︰選修
課程教師︰呂學一
開課學院：電機資訊
開課系所︰資訊工程
考試日期（年月日）︰2015/04/10
考試時限（分鐘）： 180 分鐘
試題 :
總共十題，每題十分，可按任何順序答題。只能參考個人事先準備的A4單面大抄。前八題
都是一個可能對也可能不對的敘述。如果你覺得對，請證明它是對的，如果你覺得不對，
請證明它是錯的。課堂上證過的定理，或是提過的練習題，都可以直接引用。
第一題
Let β be an orthonormal basis of a finite-dimensional inner-produt
space V over C. For any x, y ∈ V, we have
〈x∣y〉 = Σ〈x∣z〉〈z∣y〉.
z∈β
第二題
If T ∈ L(V,V) for inner-product space V over C with dim(V) < ∞, then
rank(TT*) = rank(T).
第三題
If x ∈ V and T ∈ L(V,V) for inner-product space V over C with
dim(V) < ∞, then T*(T(x)) = 0_V if and only if T(x) = 0_V.
第四題
If U and V are subspaces of inner-product space W over C with
dim(V) < ∞, then
⊥ ⊥ ⊥
(U + V) = U ∩ V
⊥ ⊥ ⊥
(U ∩ V) = U + V
第五題
If T ∈ L(V,V) for inner-product space V over R with dim(V) < ∞ such
that the characteristic polynomial f_T(t) of T splits, then T is normal
if and only if T is self-adjonit.
第六題
For any unitary T ∈ L(V,V) for inner-product space V over C with
dim(V) < ∞, there is a T' ∈ L(V,V) such that
T'T' = T
第七題
For any T ∈ L(U,V) with finite-dimensional inner-product spaces U and
V over R, there is a unique T* ∈ L(U,V) such that
〈T(x)∣y〉_V = 〈x∣T*(y)〉_U
holds for any x ∈ U and y∈V, where
●〈‧∣‧〉_U denotes the inner product of U and
●〈‧∣‧〉_V denotes the inner product of V
第八題
Matrices
┌ ┐ ┌ ┐
│i 2│ │1 4│
│ │ and │ │
│2 1│ │1 i│
└ ┘ └ ┘
are unitarily equivalent.
第九題
Find real numbers x1, x2 and x3 that satisfy the following system of
equations with x1^2 + x2^2 + x3^2 being minimized:
x1 + 5x2 = 38
x1 + 2x2 + x3 = 8
x1 - x2 + 2x3 = -22.
第十題
Find a unitary matrix Q ∈ M_3x3(C) such that
┌ ┐
│ 2 1 1 │
Q* │ 1 2 1 │ Q
│ 1 1 2 │
└ ┘
is a diagonal matrix in M_3x3(C).