课程名称︰应用代数
课程性质︰选修
课程教师︰吕学一
开课学院：电机资讯
开课系所︰资讯工程
考试日期（年月日）︰2015/05/15
考试时限（分钟）：180
试题 :
总共十一题，每题十分，可按任何顺序答题。只能参考个人事先准备的A4单面大抄。前五
题都是一个可能对也可能不对的叙述。如果你觉得对，请证明它是对的，如果你觉得不对
，请证明它是错的。课堂上证过的定理，或是提过的练习题，都可以直接引用。
第一题
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).
第二题
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.
第三题
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.
第四题
If QSR* is a singular value decomposition of a positive definite
complex square matrix, then Q = R.
第五题
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 )
第六题
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.
第七题
Let T ∈ L(V,V) for inner-product space V with dim(V) < ∞.
┴ ＋
Prove N(T) = T*(V) = T(V).
(You may directly use 奇异值定理，伪反线转定理，and 正补推论)
第八题
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.)
第九题
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│
└ ┘
第十题
Find the pseudo-inverse of the above matrix A.
第十一题
Find a polar decomposition of the above matrix A.