推 oncemore :原PO神手 01/07 23:59
課程名稱︰線性代數一
課程性質︰必修／選修
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（年月日）︰2012/1/7
考試時限（分鐘）：180
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
There are problems A to E wit a total of 140 points. Please write down
your computational or proof steps clearly on the answer sheets.
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-1
A. (28 points) Find an invertible matrix P ∈ M(4, R) so that P AP is
the Jordan form of the matrix A = ┌ 1 -1 0 -1 ┐. Write down the
│ 0 2 0 1 │
│ -2 1 -1 1 │
└ 2 -1 2 0 ┘
Jordan form of A explicitly, and find its minimal polynomial.
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B. Let A be the matrix A = ┌ 1 0 0 ┐. C(A) is the column space of
│ -1 1 0 │
│ 0 -1 1 │
└ 0 0 -1 ┘
4
A in R . (a) has 10 points, and each of (b), (c), (d), (e) has 7 points.
(a) Apply the Gram-Schmidt orthogonalization process to find an
orthonormal basis of C(A) from the column vectors of A, and then
write down the QR-decomposition of A.
T
(b) Perform the LDU-decomposition for A A to obtain a QR-decomposition
of A. Do you get the same answer as in (a)?
(c) Write down the projection matrix of the orthogonal projection from
4
R onto C(A), and find the minimal distance of the vector
T
v = [1, -1, 1, 1] to C(A).
(d) Find the least square approximate solution to the linear system
x = 3, x - x = a, x - x = 1, x = 4, where a is a constant.
1 1 2 2 3 3
What conditions should a satisfy so that this approximate solution
becomes an exact solution?
+
(e) Find A . (Moore-Penrose pseudoinverse), Use this matrix to redo (c).
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C. (14 points) Find a Schur decomposition of the matrix A =┌ 3 0 2 ┐.
│ 2 3 0 │
└ 0 2 3 ┘
Is this matrix normal?
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D. Prove the following statements. Each has 8 points.
(a) Let A ∈ M(n,F) be a nilpotent matrix with nullity δ. If its minimal
k
polynomial is x , prove that kδ >= n.
(b) Let A, B ∈ M(n,F). Prove that the set of all eigenvalues of AB equals
the set of all eigenvalues of BA, but AB and BA may not be similar.
(c) Let A ∈ M(n,C). Prove that A is unitary iff A is normal, and every
eigenvalue of A has absolute value 1.
(d) Let V be a finite-dimensional inner product space over F, Its inner
product is denoted by <u, v> for u,v ∈ V. If W ⊂ V is a subspace,
┴
prove that W is isomorphic to V/W.
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E. Determine which of the following statements is true. Prove your answer.
Each has 7 points.
(a) Let A, B ∈ M(n,R). Then they are similar in M(n,C) iff they are
similar in M(n,R).
(b) Let A, B ∈ M(3,F). Then A and B are similar iff whenever A satisfies
p(A) = 0 for some polynomial p(x), B must satisfy p(B) = 0, and
vice versa.
+
(c) Given A ∈ M(n,C) with det(A) = 0. Then the nullities of A and A
+ +
are the same, and AA = A A.
2
(d) For a given matrix A ∈ M(n,F), the number of Jordan blocks of A
is the same as the number of Jordan blocks of A.
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※ 編輯: arsenefrog 來自: 140.112.239.2 (01/07 23:28)