2018年华南理工大学864高等代数考研真题试题试卷
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5. (20) A P V , f (x) ∈ P [x]f (A) = 0, f (x) = f1 (x)f2 (x) · · · fs (x) f1 (x), f2 (x), ..., fs (x) .
Vi = (fi (A))−1 (0), i = 1, 2, ..., s. V = V 1 ⊕ V2 ⊕ · · · ⊕ V s .
...让知识更美味...
864
2018
, , , , 2
(x) 1. (20) f (x), g (x) ∈ P [x], d(x) = (f (x), g (x)) ∂ ( f ) ≥ 1, d(x) g (x) ) ≥ 1, u(x), v (x) ∈ P [x] ∂( d (x)
u(x)f (x) + v (x)g (x) = d(x),
∂ (u(x)) < ∂ ( d(x) ), ∂ (v (x)) < ∂ ( d(x) ).
g (x) f (x)
2. (20) n: 1 a1 Dn = 1 a2 a2 1 a2 2
−2 −1 · · · an aLeabharlann Baidu + 1 1 −2 −1 · · · an an + 2 2 S a1 S a2
6. (15) ⎧ ⎨ x1 + x3 = 0, ⎩ x +x =0 2 4
W , α = (2, 3, 4, 5)W αW .
(1)
(: V = V1 ⊕ V1⊥ , α ∈ V α = α1 + α2 , α1 ∈ V1 , α2 ∈ V1⊥ , α1 αV1 .)
··· ···
···
S an
,
n−2 −1 1 a n a2 an + n · · · an n
S =
n i=1
ai . ⎛ 1 1 1 1 ⎞
⎜ ⎟ ⎜ ⎟ ⎜ 1 1 −1 −1 ⎟ ⎜ ⎟. 3. (20) A = ⎜ ⎟ ⎜ 1 −1 1 −1 ⎟ ⎝ ⎠ 1 −1 −1 1 −1 (1) A ; (2) AB − A = B , B AB −1 .
2
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2018 年华南理工大学 864 高等代数考研真题试题试卷·············································· 2
第 1 页，共 3 页
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第 2 页，共 3 页
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4. (20) Am × s, B m × n. (1) AX = B
r (A, B ) = r (A); (2) XA = B
2 2 7. (15) f (x1 , x2 , x3 ) = x2 1 + x2 + x3 − 4x1 x2 − 4x1 x3 + 2 2 2 + 3 y2 + by3 ,a, b 2ax2 x3 x = P y f = 3y1
P .
8. (20) An × n, k , Ak = 0, : An = 0.
Vi = (fi (A))−1 (0), i = 1, 2, ..., s. V = V 1 ⊕ V2 ⊕ · · · ⊕ V s .
...让知识更美味...
864
2018
, , , , 2
(x) 1. (20) f (x), g (x) ∈ P [x], d(x) = (f (x), g (x)) ∂ ( f ) ≥ 1, d(x) g (x) ) ≥ 1, u(x), v (x) ∈ P [x] ∂( d (x)
u(x)f (x) + v (x)g (x) = d(x),
∂ (u(x)) < ∂ ( d(x) ), ∂ (v (x)) < ∂ ( d(x) ).
g (x) f (x)
2. (20) n: 1 a1 Dn = 1 a2 a2 1 a2 2
−2 −1 · · · an aLeabharlann Baidu + 1 1 −2 −1 · · · an an + 2 2 S a1 S a2
6. (15) ⎧ ⎨ x1 + x3 = 0, ⎩ x +x =0 2 4
W , α = (2, 3, 4, 5)W αW .
(1)
(: V = V1 ⊕ V1⊥ , α ∈ V α = α1 + α2 , α1 ∈ V1 , α2 ∈ V1⊥ , α1 αV1 .)
··· ···
···
S an
,
n−2 −1 1 a n a2 an + n · · · an n
S =
n i=1
ai . ⎛ 1 1 1 1 ⎞
⎜ ⎟ ⎜ ⎟ ⎜ 1 1 −1 −1 ⎟ ⎜ ⎟. 3. (20) A = ⎜ ⎟ ⎜ 1 −1 1 −1 ⎟ ⎝ ⎠ 1 −1 −1 1 −1 (1) A ; (2) AB − A = B , B AB −1 .
2
第 3 页，共 3 页
芝士传媒 × 题源库
www.zhishi360.com
...让知识更美味...
目 录
2018 年华南理工大学 864 高等代数考研真题试题试卷·············································· 2
第 1 页，共 3 页
芝士传媒 × 题源库
www.zhishi360.com
1
第 2 页，共 3 页
芝士传媒 × 题源库
www.zhishi360.com
...让知识更美味...
4. (20) Am × s, B m × n. (1) AX = B
r (A, B ) = r (A); (2) XA = B
2 2 7. (15) f (x1 , x2 , x3 ) = x2 1 + x2 + x3 − 4x1 x2 − 4x1 x3 + 2 2 2 + 3 y2 + by3 ,a, b 2ax2 x3 x = P y f = 3y1
P .
8. (20) An × n, k , Ak = 0, : An = 0.