課程名稱：線性代數一
課程性質︰數學系必修
課程教師︰林紹雄
開課學院：理學院
開課系所︰數學系
考試日期︰2008年01月13日
考試時限：180分鐘
是否需發放獎勵金：是
試題 :
Math 201 14410(Linear Algebra) Final
There are problem A to F with a total of 130 points. Please write down your
computational or proof clearlyon the answer sheets.
A. Let A be the matrix
┌ 1 -1 0┐
｜-1 1 -1｜
A= ｜ 0 -1 1｜
└ 0 0 -1┘
Let V be the column space of A in R^4. Each has 7 points in the following
problems.
(a) Use the Gram-schmidt orthogonalization process to find an orthonormal
basis of V from the column vectors of A,and write down the
QR-decomposition of A
(b) Perform the LDU-decomposition for (A^T)A to obtain the QR-decomposition
of A.Do you get the same answer as in (a).
(c) Write down the projection matrix of the orthogonalprojection from R^4
to V,and find the minimal distance from the vector v=[ 1,-1,1,-1]^T
to V
(d) Find the best least square approximation solution to the linear system
x1-x2=3, x1-x2+x3="a",x2-x3=-1 ,x3=4
where"a"is a constant.What conditions should "a" satisfy so that this
approximate solution becomes an exact solution?
B.(15 points) Find the charateristic polynomials,eigenvalues and the
eigenvectors of the following matrix
┌ 5 -6 -6┐
A= ｜-1 4 2｜
└ 3 -6 -4┘
Determine whether it is diagonalizable. If yes,find an invertible P
such that P^-1AP is diagonal. Is this matrix a normal matrix?
C.Let A be the matrix
┌ -1 1 0 0 ┐
｜ -1 0 1 0 ｜
A= ｜ 0 1 0 -1 ｜
└ 0 0 -1 1 ┘
Each has 5 points in the following problems.
(a) Draw a digraph with numbered and directed edges (and numbered nodes)
such that its incidence matrix is A
(b) Write down the Kirchhoff Voltage Law to determine when a vector f belong
R^4 lies in the column space of A
(c) How many independent loops exist in the graph?
(d) Is this graph a tree? show that removing the last edge produce a
spanning tree.
D.(12 points) Find a Schur decomposition of the following matrix
┌ 3 0 2 ┐
A= ｜ 2 3 0 ｜
└ 0 2 3 ┘
E. Prove the following statement. Each has 10 points.
(a) Let A and B be m*n and n*m matrices respectively. Prove that
┌ 0 A┐
det└-B I┘ = det(AB)
Moreover,if m>n, prove that det(AB)=0
(b) Let A belong to Mn(C). Prove that A is normal iff there exists a unitary
matrix U belong to Mn(C) such that A*=AU
(c) If A is a nonsingular square matrix ,prove that there exists a
permutation matrix P such that PA has no zeros on its main diagonal
(d) Let A and B be m*n and n*m real matrices respectively .Assume that
A has independent column vectors, ABA=A ,BAB=B,(AB)^T=AB,(BA)^T=BA
Prove that B=(A^TA)^-1A^T
F. Determine which of the following statements is true.Prove your answer,
or give a counterexample. Each has 5 points.
(a) There exists a 3*3 matrix A each of whose term is 1 or -1 such that
det(A)=5
(b) Let P be the transposition matrix obtained by exchanging the first two
rows of In,Then adj(PA)=-adj(A)P for a matrix A belong to Mn(R)
(c) Let P be a 3*3 permutation matrix so that det(P)=-1,Then there exists
a family of intertible matrices A(t) belong to M3(R) for 0≦t≦1 such
that A(t) is continuous in t, and A(0)=-I3,A(1)=P
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