华南理工大学高等代数 2020考研真题

100
A
=
1
0
1
.
010
1. y²: An = An−2 + A2 − E (n ≥ 3). 2. ¦ A2020.
o. ( 20 ©) ®• α1 = (1, 2, 0), α2 = (1, a + 2, −3a), α3 = (−1, −b − 2, a + 2b), β = (1, 3, −3), ¦ a, b Š ¦
1. β ØŒ α1, α2, α3 ‚5LÑ.
2. β Œ α1, α2, α3 •˜‚5LÑ, ¿¦Lˆª.
3. β Œ α1, α2, α3 ‚5LÑ…Ø•˜, ¿¦Lˆª.
nn
Ê. ( 20 ©) ®• f (x1, x2, · · · , xn) =
aijxixj, K.5•ê©O• p, q, … α1, α2, · · · , αp, β1, β2, · · · , βq
. ( 15 ©) OŽ n + 1 1 ª
an (a − 1)n (a − 2)n · · · (a − n)n
an−1 (a − 1)n−1 (a − 2)n−1 · · · (a − n)n−1
D = ...
...
...
...
.
a
a−1
a−2 ··· a−n
1
1
1
···
1
n. ( 20 ©) ®•
18. uHnóŒÆ 2020 cp “ê考研真题
˜. ( 20 ©) ®• f (x), g(x) ∈ P [x], y²: (f (x), g(x)) = 1 ¿©7‡^‡´é ∀r(x), s(x), ∃q(x), p(x)
¦ p(x)f (x) + r(x) = q(x)g(x) + s(x).
i=1 j=1
•?¿ p + q ‡ ê. y²: •3šòz‚5O† X = CY ¦ f (X) = α1y12 + · · · + αpyp2 − β1yp2+1 −
· · · − βqyp2+q.
8. ( 20 ©) V ´ê• P þ n ‘‚5˜m, … V = P [x]n, k A (f (x)) = xf (x) − f (x), f (x) ∈ V .
Baidu Nhomakorabea
l. ( 15 ©) ®• V ´ê• P þ Ý•
n ‘‚5˜m, ε1, ε2, ε3, ε4 ´ V
1 −1 −1 2
A
=
0
1
0
0
.
2 3 1 1
1 −2 −2 −1
¦•¹ ε1 • ØCf˜m.
˜|Ä, A 3 ε1, ε2, ε3, ε4 e
1. y²: A •‚5C†. 2. ¦ A −1(0) † A (V ). 3. y²: V = A −1(0) ⊕ A (V ).
Ô. ( 20 ©) ®• A, B •ê• P þ n • , A k n ‡pÉ A Š, y²: A A •þ´ B A •þ ¿©7‡^‡´ AB = BA.