課程名稱︰線性代數二
課程性質︰數學系必修
課程教師︰陳其誠
開課學院：理學院
開課系所︰數學系
考試日期︰2011年04月29日
考試時限：110分鐘
是否需發放獎勵金：是
試題 :
LINEAR ALGEBRA EXAM II
4/29 2011
Write your answer on the answer sheet.
┌-1 0 2┐
(1) (25 points) Let A =│ 0 1 2│. Find an orthogonal matrix Q and a diagonal
└ 2 2 0┘
matrix D so that (Q)A(Q^t) = D.
(2) (20 points) Find the area of the region enclosed by the ellipse:
89x^2 + 96xy + 61y^2 + 170x + 190y + 25 = 0.
(3) Either give a brief reason or give a counter example for each of the foll-
owing assertions (5 points each):
(a) Suppose T is a normal operator on a finite dimensional complex inner
product space. Then every eigenvector of T is also an eigenvector of
T*.
(b) Suppose T is a linear operator on a finite dimensional complex inner
product space such that every eigenvector of T is also an eigenvector
of T*. Then T must be normal.
(c) The eigenvalues of a self-adjoint operator must be real.
(d) Every orthogonal projection is a unitary operator.
(e) If T and U are unitary operators on an inner product space, then so is
TU.
(4) Give a rigorous proof for each of the following assertions.
(a) (18 points) Let T be a linear operator on a finite dimensional real
inner product space V. Then T is self-adjoint if and only if there ex-
ists an orthonormal basis β for V consisting of eigenvectors of T.
(b) (10 points) Let T be a normal operator on a finite dimensional inner
product space. If T is a projection, then T must be an orthogonal pro-
jection.
(c) (2 points) Suppose V is an inner product space and T:V->V is a map. If
||T(x)|| = ||x|| holds for every x∈V, then <T(x),T(y)> = <x,y>.
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