課程名稱︰線性代數二
課程性質︰數學系大一必修
課程教師︰莊武諺老師
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2019/4/17
考試時限（分鐘）：150
試題 :
(以下的ε當作屬於的符號來使用)
(T* denotes the adjoint of a linear transformation T)
(There are totally 110 points)
(1) (15 points) Let A,BεMnxn(R) (nxn real matrices). Show that A+iB is unitary
if and only if (A -B) εM2nx2n(R) (2nx2n real matrices) is
orthogonal. B A
(2) (15 points) Let V be the vector space over C, consisting of all the complex
sequences (a1,a2,a3,...) with only finitely many nonzero
entries. The inner product on V is given by
〈(a1,a2,a3,...),(b1,b2,b3,...)〉=sigma(i from 1 to infinity)
aibi (bi takes complex conjugate)
Let ei be the sequence with only one nonzero entry being 1 at
i-th position. Let T: V→V be the linear transformation
defined by T(en)=ne(n+1)+(n+1)e(n+2)+(n+2)e(n+3) for nεN.
Does T has an adjoint? If yes, please write it down.
(3) (15 points) Let (V,〈-,-〉) be an inner product space over R. Let W be a
subspace of V. Suppose that the orthogonal projection
T=ProjW: V→V onto W exists, i.e.,T^2=T,N(T)=R(T)⊥, and
R(T)=N(T)⊥.
(i) Show that T is self-adjoint.
(ii) Find a self-adjoint operator S: V→V such that S^3=1+2T.
(4) (15 points) A linear operator on a finite-dimensional inner product space
is called positive semidefinite if T is self-adjoint and
〈T(x),x〉≧0 for all nonzero xεV. Show that T is positive
semidefinite if and only if T=B*B for some B: V→V.
(5) (15 points)
(i) Let a1,a2,...ak be k distinct complex numbers. Show that there exists a
polynomial g(t)εC[t] with complex coefficients such that g(ai)=bi for
any given biεC, i=1,2,...,k.
(ii) Let (V,〈-,-〉) be a finite-dimensional complex inner product space
and T be a linear operator on V. Show that T is normal if and only if
T*=g(T) for some polynomial g(t)εC[t].
(6) (20 points) Let (U,〈-,-〉U), and (V,〈-,-〉V) be finite-dimensional inner
product spaces over F.
(i) Let S: U→V be a linear transformation. Show that there exists a unique
linear transformation S*: V→U such that 〈S(x),y〉V=〈x,S*(y)〉U.
(ii) Show that R(S*)=N(S)⊥.
(7) (15 points) Let (U,〈-,-〉U), (V,〈-,-〉V), and (W,〈-,-〉W) be
finite-dimensional inner product spaces over R. Let
S: U→V and T: V→W be linear transformation such that
R(S)=N(T). Define the linear operator L(x,y):= x^2SS*+y^2T*T
for x,yεR. Please describe the function
d(x,y):=nullity(L(x,y)) on R^2.