課程名稱︰ 工程數學-線性代數
課程性質︰
課程教師︰ 馮蟻剛
開課系所︰ 電機系
考試時間︰ 2006/6/21
ps.以下，e^2 表示上標；P_v 表示下標；向量u以vec(u)表示，以此類推。
試題 :
1. Judge if the following ststement are true or false. Give a concise proof
to each true statement, and a counterexample to each false statement.(25%)
(a) For vec(u) and vec(v) in R^n, <vec(u)‧vec(v)>=(A vec(u))‧vec(v) defines
an inner product on R^n with a given n x n invertible matrix A.
(b) For any two subspaces V and W of R^n , V is contained in W implies
that (W perp) is contained in (V perp).
(c) For any two subspaces V and W of R^n , P_v + P_w is an orthogonal
projection matrix, where P_v and P_w are the orthogonal projection
matrices for V and W, respectively.
(d) If A is a symmetric matrix, then its largest singular value equals its
largest eigen value.
(e) For any n x n invertible matrix A, A^+ = A^(-1).
2. For V = Span{e^t , te^t , t^2 e^t} and the differential operator D, find
all eigenvector for each eigenvalue. (15%)
3. For any n x n matrix A, the minimal polynomial of A is defined as the
polynomial p(t)= t^m + p_(m-1)t^(m-1) +...+ p_1 t + p_0 with the lowest
degree m such that p(A)=A^m + p_(m-1)A^(m-1) +...+ p_1 A + p_0 I_n = O.
Find the minimal polymials of the following matrices and justify your
answers.
(a) The elementary matrix E obtained form I_n by interchanging its ith and
jth rows (r != j). (10%)
(b) The n x n matrix M with k distinct eigenvalues λ_1,λ_2,...,λ_k,
where k <= n and λ_i has the algebraic multiplicity n_i for
i=1,2,...,k. (10%)
4. Find all eigenvalues and their corresponding eigenspaces of the matrix E
in problen 3(a). (10%)
5. Given the SVD of
A=[0 1 2]=[ 2/√5 -1/√5] [√6 0 0] [1/√30 2/√5 1/√6]^T
[1 0 1] [ 1/√5 2/√5] [ 0 1 0] [2/√30 -1/√5 2/√6]
[5/√30 0 -1/√6] ,
plot the image of the unit sphere
{(x_1,x_2,x_3)│(x_1)^2 + (x_2)^2 + (x_3)^2 =1,x_i belongs to R for i=1,2,3}
in R^3 under the mapping of T_A. (15%)
6. A linear operator T on a finite-dimensional inner product space V is called
an orthogonal operator if [T]_B is an orthonorgonal matrix for some basis B
of V. Prove that T is an orthogonal operator if and only if for any
orthonormal basis
{vec(v_1) , vec(v_2),...,vec(v_n)} the set
{T(vec(v_1)) ,T(vec(v_2)),...,T(vec(v_n))} is also an orthonormal basis for
V. (15%)
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