課程名稱︰線性代數二
課程性質︰數學系大一必修
課程教師︰謝銘倫
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2016/06/15
考試時限（分鐘）：140
試題 :
Part A
t
Problem 1 (20 pts). Find an orthogonal matrix P such that P AP is diagonal,
where
╭ 4 2 2 ╮
│ │
A = │ 2 4 2 │.
│ │
╰ 2 2 4 ╯
Problem 2 (20 pts). Let
╭-3 0 ╮
│ │
A = │ 3 -3 │ ∈ M (|R).
│ │ 3×2
╰ 0 3 ╯
(1) Find the singular value decomposition (S.V.D.) of A and the Moore-Penrose
+
inverse A of A.
(2) Show that there is no solution to the following linear equation
╭ 1 ╮
│ │ 2
Ax = │ 2 │, x ∈ |R .
│ │
╰ 3 ╯
Find the best approximate solution to the above equation.
Problem 3 (20 pts). Let
╭ 1/4 1/4 1/2 ╮ ╭ 1 ╮
│ │ │ │
A = │ 1/4 1/4 1/4 │ and w = │ 1 │.
│ │ │ │
╰ 1/2 1/2 1/4 ╯ ╰ 1 ╯
(1) Find the Perron-Frobenius vector v of A.
A
n
(2) Find lim A w.
n→∞
Problem 4 (25 pts). Let V = M (|R) be a four-dimensional vector space over |R.
2
╭ 1 2 ╮ ╭ 1 0 ╮
Let A = │ │ and B = │ │. Define the linear transformation L: V →
╰ 0 1 ╯ ╰ 3 1 ╯
V by
L(X) = AXB.
(1) (10 pts) Choose your favorite basis \mathscr{A} of V. Write down the matrix
[L] ∈ M (|R) for L with respect to this basis \mathscr{A}.
\mathscr{A} 4
(2) (10 pts) Determine the Jordan canonical form of L.
(3) (5 pts) Let Q: V → |R be the quadratic form defined by Q(X) := det X.
Determine the signature of Q.
Problem 5 (15 pts). Let T: V → V be an isometry for the quadratic space (V, Q)
in the previous problem. Show that there exist invertible matrices C, D ∈
t
M (|R) such that either T(X) = CXD or T(X) = CX D.
2
Part B
(Caution: No partial credit for bonus problems)
Problem 6 (10 pts). Let A, B, C ∈ M (|R). If ABC = 0 and rank B = 1, show that
n n
either AB = 0 or BC = 0 .
n n
p+1
Problem 7 (10 pts). Let A ∈ M (|R). Suppose that A = A, where p is a prime
n
number. Show that
2 p-1
rank(A - I ) = rank(A - I ) = ... = rank(A - I ).
n n n
Problem 8 (10 pts). Is it possible to find polynomials a (x), a (x), a (x) ∈
1 2 3
|C[x] and b (y), b (y), b (y) ∈ |C[y] such that
1 2 3
2 2 3 3
1 - x - y + 2xy + x y + x y = a (x) b (y) + a (x) b (y) + a (x) b (y)?
1 1 2 2 3 3
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