課程名稱︰線性代數
課程性質︰系定必修
課程教師︰馮世邁、逢蟻剛、黃升龍、劉志文
開課學院：電資學院
開課系所︰電機系
考試日期（年月日）︰2007.6.20
考試時限（分鐘）：100
是否需發放獎勵金：是，謝謝
（如未明確表示，則不予發放）
試題 :
1. Consider the equation X^2=I2, where X is a 2*2 matrix and I2 is the 2*2
identity matrix.
(a) Show that every solution matrix X is diagonalizable.(10%)
(b) Find the set of all solution matrices X.(10%)
2. Consider an SVD of A,
┌0 1┐ ┌1/√30 2/√5 1/√6┐┌√6 0┐ T
│ │ │ ││ │┌2/√5 -1√5┐
A= │1 0│= │2/√30 -1/√5 2/√6││ 0 1││ │
│ │ │ ││ │└1/√5 2√5┘
└2 1┘ └5/√30 0 -1/√6┘└ 0 0┘
show that the image of the unit disk
{(x1,x2)|x1^2+x2^2≦1, xi(屬於)R for i=1,2.}
in R2 under the mapping of TA is an ellipse. Find the direction and length
of the long axis of the ellipse. (10%)
3. Let F:V→V be a linear operator on the vector space
T T
V=Span{[2 0 -1] ,[-3 1 4] }.
T T T T
Suppose F([2 0 -1] )=[-1 1 3] and F([-3 1 4] )=[5 -1 -5] .
T T
(a) With a basis B ={[-1 1 3] , [5 -1 -5] } of V, find [F]B and use it
T
(required!) to find the vector F^-1([2 2 4] ).(10%)
(b) Find the eigenvalues and corresponding eigenvectors of F. (10%)
4.(a)If λ is an eigenvalue of A with associated eigenvector v, show that λ^k
is an eigenvalue of A^k with associated eigenvector v, where k is a
positive integer. (10%)
(b)If A^k=O for some positive integer k, show that the only eigenvalue of A
is 0. A is an n*n matrix. (15%)
5.(a)Describe all vector that are orthogonal to the null space of the matrix A.
┌1 3 7┐
│ │
A=│2 2 6│.
│ │
└2 1 4┘
(b)Apply Gram-Schmidt process to the columns of A to obtain an orthonormal
set.(15%)
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