課程名稱︰線性代數二 課程性質︰數學系大一必修 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（年月日）︰2011/04/29 考試時限（分鐘）：110分鐘 是否需發放獎勵金：是 （如未明確表示，則不予發放） 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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