課程名稱：工程數學--線性代數
課程性質︰
課程教師：馮世邁
開課系所︰電機系
考試時間︰2006.6.21
試題 :
1.Let A be a m*n matrix.Prove that rank (A^T)*A = rank A (10%)
2.Suppose that T:R^2 -> R^2 is an orthogonal operator.Let U:R^2 -> R^2 be
defined by
U( [x1] ) = [1/(根號2)]*T( [x1] )
[x2] [x2]
(a)Prove that U is an orthogonal operator. (5%)
(b)If λ is an eigenvalue of an orthogonal matrix Q, prove that λ
is either 1 or -1. (5%)
(c)Prove that if there is an orthonormal basis for R^n consisting of
eigenvectors of A, then A is symmetric. (5%)
3.Consider the recursive equation R(n)=aR(n-1)+c where a and c are
constants and a≠1. Let
Sn=[ 1 ] and A=[1 0]
[R(n)] [c a]
(a)Prove that there exist eigenvectors v1 and v2 of A and scalars b1 and b2
such that
So = b1v1+b2v2
Give a pair of v1 and v2 and obtain the correstponding b1 and b2
(as functions of a,c and R(0) )
(b)Express R(n) in terms of a, c,and R(0).
4.Let P3 be the inner product space consisting of polynomials of order ≦3
with the inner product defined by
1
<p1(x),p2(x)> = ∫ p1(x)*p2(x)*dx
-1
Let V be the subspace of P3 defined by
V={a0+a1*x+a2*x^2+a3*x^3 | a2+a3=0}
(a)Find a basis for V. (3%)
(b)Find a orthonormal basis B for V.(7%)
(c)Find the least squares approximation to the function f(x)=三次根號x, with
domain restricted to [-1,1], as a polynomial in V. (8%)
(d)Let T: V -> V be the linear operator defined by T(p(x)) = p(x)+p(-x). Find
the matrix representation [T]B where B is given in (b).
(本題因為T並不是V映到V，所以送分) (8%)
(e)Find a basis for Null T and Range T. (送分)(5%)
5.(a)Find the singular value decomposition of the following matrix.(10%)
[0 1]
A= [1 0]
[1 1]
(b)Let B be an m*n matrix. Show that Bt*(Bt*B+Im)^(-1) = (B*Bt+In)^(-1)*Bt
(Hint: singular value decomposition of B)
(Bt = B trasnpose) (10%)
6.Prove that, for any matrix A, A*(A#) is the orthogonal projection matrix
for ColA, and (A#)*A is the orthogonal projection matrix for Row A.
(The matrix A# is the Moore-Penrose generalized inverse or psudoinverse
of A)
(10%)
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