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）︰2015/05/15
考試時限（分鐘）：180
試題 :
總共十一題，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。前五
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 V be an inner-product space over C with dim(V) < ∞. If T is a
projection of V with TT* = T*T, then T is the orthogonal projection
of V on T(V).
第二題
Let V be an inner-product spave over R with dim(V) < ∞. If T ∈ L(V,V)
is self-adjoint, then there is a normal T' ∈ L(V,V) with T'T' = T.
第三題
Let V be an inner-product space over C with dim(V) < ∞. If T is a
projection of V with T* = T, then each eigenvalue of T is either 0_R
or 1_R.
第四題
If QSR* is a singular value decomposition of a positive definite
complex square matrix, then Q = R.
第五題
If A is an m ×n real matrix and B is an n ×n orthogonal real matrix,
＋ ＋＋
then (AB) = B A.
t -1
(Recall that B is orthogonal if B = B )
第六題
Let T1, T2 ∈ L(V,V) for inner-product space V with dim(V) < ∞.
Prove that if T1T2T1 = T1, T2T1T2 = T2, (T1T2)* = T1T2, and
＋
(T2T1)* = T2T1, then T1 = T2.
第七題
Let T ∈ L(V,V) for inner-product space V with dim(V) < ∞.
┴ ＋
Prove N(T) = T*(V) = T(V).
(You may directly use 奇異值定理，偽反線轉定理，and 正補推論)
第八題
Let T be a linear operator on a finite-dimensional vector space. Prove
that T is a projection if and only if TT = T. (You may directly use
properties of direct sum shown in class.)
第九題
Find a singular value decomposition of
┌ ┐
│1 0 0 0 2│
│0 0 3 0 0│
A = │0 0 0 0 0│
│0 4 0 0 0│
│0 0 0 0 0│
└ ┘
第十題
Find the pseudo-inverse of the above matrix A.
第十一題
Find a polar decomposition of the above matrix A.