复旦大学高等代数试题

A
n×n
φ(α) = Aα ) R3 )
φ
α1 = (1, 2, 2), α2 = (5, −2, 4) , α3 = (6, −3, 0) α1 , α2 , α3 ( V
∗ ∗ ∗ v1 , v2 , v3
F
(
V∗ V )
v1 , v2 , v3 V v1 , v1 − v2 , v1 + v2 − v3
2006∼2007
√
A II
B MATH30002.01
1.
2.
A
2 1
1 1 t 0 2 A A=
0 t 2 1

1, −1, 2
A2 −2A−1 +I
(
) 2 1 0
1 1 1− t 2 t 1 0 2 A16 = I A=
2A
f (x, y, z ) =
2x2 + y 2 − 4xy − 4yz ( z2 + y2 + z2
x2 + y 2 + z 2 = 0)
α α
V1 V1
V = V1 ⊕ V1⊥ V1
α = β + γ, α ∈ V, β ∈ V1 , r ∈ V1⊥ ξ ∈ V1
β β ∈ V1
α−β ≤ α−ξ
3. 4.
5 −6 3 −4
A10 =
5.
6. 7.
−1 −2 6 A = −1 0 3 −1 −1 4 1 1 0 A= 0 2 1 0 0 2 A 3×3
A
(
Baidu Nhomakorabea
)
A −1, −2, −3
( A A∗
) ( )
1A
8.
V
n
V (α, β ) = α′ Gβ,
G A 9. 10. (
3A
A 1. 2. 3. T
n n T
T (X ) = XA + A′ X Rn×n
A3 = −I A −A
T
f (λ) = an λn + an−1 λn−1 + · · · + a1 λ + a0 Q A f (A) = 0 1. 2. A A Q C
Q
1
4A
3 R[x]3 , (f (x), g (x)) = 1. 2.
R[x]3
1 −1
f (x)g (x)dx R[x]3
∀f (x), g (x) ∈
(f (x), g (x)) R[x]3
h1 (x) = 1, h2 (x) = x − 1, h3 (x) = x2
f (x1 , x2 , · · · , xn ) = (x1 + a1 x2 )2 +(x2 + a2 x3 )2 + · · ·+(xn−1 + an−1 xn )2 +(xn + an x1 )2 f