09年第一学期期末试卷及答案.pdf

....................... .......................................... .......................................... ....................................... ) ( C ¾ ‚ S Ø ‡ ‰ K E ŒÆêƉÆÆ 2008*2009Æc1ÆÏÏ"ÁÁò Aò ‘§¶¡µ p“êI m‘Xµ êƉÆÆ 6 ¶µ Æ Òµ K Ò 1 2 3 ‘§ S Òµ MATH120011 Á/ªµ 4ò ; ’µ 4 5 6 7 o 8  © © ˜ !À JK (z zK 2.5© ©§
20© ©) 1. ˜‡ n (n > 1)  ( D ¥z˜‘Šþ 1 ½ −1§KTÝ ) A. 1 a11 a12 a13 a21 a22 a23 = d§K a31 a32 a33 A. 3d ( A A. 1 −6a22 3a23 2a31 −4a32 2a33 −a11 2a12 −a13 § =( C ) C. 12d D. 18d −1 A−1 ! B −1 ! (A + B)−1 þ3"K (A−1 + B −1 ) ) A. A (A + B)−1 B 4.  t = ( 3a21 B. 6d 3.  A ! B Ó D. óê C. Ûê B. 0 2. e1ª 1ª C B. A (A + B)−1 C. (A + B)−1 B D. A + B ) ž§•þ| (1, 1, 1)§(t, 1, 2)§(1, t2 , 1) ‚5Ã'º B. −1 1 1  (
8 ) C. −2 D. 2 = 5.  V ê K þ n (n > 1) ‘‚5˜m"e(Ø ( C ) é V þ¤ k‚5C† ϕ Ѥá" A. V = Im ϕ ⊕ Ker ϕB. dim V = dim Im ϕ + dim Ker ϕC. V = Im ϕ + Ker ϕD. Im ϕ ∩ Ker ϕ = {0} 6.  p (x) ´ê K þ،õ‘ª§f (x) = pm (x) (m > 1)§g (x)§h (x) ∈ K [x]"Ke(Ø ( A ) (" A. f (x) | g (x) h (x) ⇒ f (x) | g (x) ½ f (x) | h (x) B. f (x) | g (x) h (x) ⇒ f (x) | g (x) ½3ê k§f (x) | hk (x) C. (f (x) , f 0 (x)) 6= 1 D. f (x) 3 C þk­Š" 7.  a, b ©O ( A. −1§2 D ) ž§(x − 1)2 Ø ax4 + bx2 + 1" B. −1§−2 C. 1§2 D. 1§−2 8.  A§B Ñ´ê K þ n (n > 1) Ý "Ke(ؤá´ ( D ) A. e Ax = 0 )Ñ´ Bx = 0 )§K A •þÑ´ B •þ‚ 5|ܶ B. e Ax = 0 )Ñ´ Bx = 0 )§K B •þÑ´ A •þ‚ 5|ܶ C. e Ax = 0 )Ñ´ Bx = 0 )§K A 1•þÑ´ B 1•þ‚ 5|ܶ D. e Ax = 0 )Ñ´ Bx = 0 )§K B 1•þÑ´ A 1•þ‚ 5|ܶ  !W ˜K (z zK 2.5© ©§
20© ©) 1. ü‡õ‘ª f (x) = x4 − 2x3 − 3x2 + 9x − 6 Ú g(x) = x3 − 6x2 + 12x − 8  ŒúϪ (f (x), g(x)) = x−2 . 1 2  (
8 ) .................... .......................................... .......................................... .......................................... ) ( C ¾ ‚ S Ø ‡ ‰ K 2. ®Ý X ÷ve§µ     1 2 3   1 2 1   , X  −1 4 3  =    −1 3 1 2 0 1   K X = 5 − 3 3 − 2 4 2 3 3 1 2   .  3. ,‚5˜m V þ‚5C† ϕ 3,|ÄeL«Ý K ϕ ؘm‘ê dim Ker(ϕ) = 1 1    3 A=   0  4 1 1 1    2 1 1  §  1 2 2   5 3 3 . 4.  A  n  §A∗ L« A Š‘Ý §® |A∗ | = 2n−1 , K |A−(A∗ )∗ | = 2(ω − 2n−2 )n Ù¥ ω n−1 = 1 . 5. X J Km [x] L « ê  K þ g ê Ø ‡ L m  ˜  õ ‘ ª | ¤  ‚ 5 ˜ m §  U = K3 [x], V = K4 [x]§ y k ˜ ‡ ‚ 5 N  T : U 7→ V ÷ v Rx T (f ) = 1 f (t)dt, ∀f (x) ∈ U . © O  ½ U , V  Ä {1, x + 1, x2 , x3 } Ú {1, x, x2 , (x + 1)3 , x4 }§ KU‘þ½Âª§T 3ùü|Äe éAÝ A 3 2 1  −1 − − −  2 3 4     1  1 −1 0     1  0  −1 0   2   1  0  0 0   3   1 0 0 0 4  A= . 6. õ‘ª xp + px + 1 £p Ûƒê¤3knêþ£´/Ĥ 1 3  (
8 ) Ä Œ"  7.  M = A B O C  M −1 = A   , Ù¥ A, C þŒ_ §K −1 −1 −A BC O C −1 −1   . 8. ® V ´˜‡ 3 ‘E‚5˜m§e 1 , e2 , e3 ´˜|ħV þ‚5C† ϕ 3ù   0 −a   |ÄeÝ  A =  1 2a  0 0 f˜m§ K a Š‰Œ 0   V Tk 3 ‡p؃Ó 1-‘ ϕ-ØC 0   1 a 6= 0Úa 6= 1 . n !( K 10© ©)  V ´•þ˜m R3 , e1 , e2 , e3 ´ V Ä•þ|§ ϕ´  2 −1 3     V þ‚5C† ϕ 3Ä•þ|eÝ  A =  1 2 0 . yk•þ   −2 2 1 t t t | α1 = (−1, −1, 2) §α2 = (3, 2, 0) , α3 = (−1, −1, 1) ∈ V . ¯ (1) α1 , α2 , α3 ´Ä V Ä? (2) XJ α1 , α2 , α3 ´ V ħ¦ ϕ 3ù|ÄeÝ "   −1 3 −1     (α1 , α2 , α3 ) = (e1 , e2 , e3 )  −1 2 −1    2 0 1 −1 3 −1 1. Ï  −1 2 −1 2 § 0 1   −1 3 −1     = −1 6= 0§ ¤ ± −1 2 −1 ´ Œ _ Ý   2 0 1 α1 , α2 , α3 ´ V Ķ 1 4  (
8 )   −1 3 −1     2. PP =  −1 2 −1 §K(α1 , α2 , α3 ) = (e1 , e2 , e3 ) P §ϕ (α1 , α2 , α3 ) =   2 0 1 −1 ϕ (e1 , e2 , e3 ) P = (e1 , e2 , e3 ) AP = (α1 , α2 , α3 ) P  AP §ϕ3ù |ÄeÝ −17 11 −12     −1 P AP =  8 −3 5 "   36 −24 25 o ! ( K ©) ® α1 = (2, 1, 4, −2)t §α2 = (5, −1, 3, 3)t Ú α3 = (1, 0, 1, 0)t  10©   2x1 − 9x2 − cx3 + bx4 = 1      x − 11x + ax + bx = 1 1 2 3 4 n‡)§ ´§|   5x − 29x − 2x − 15x = 3 1 2 3 4      4x − 31x − x − 15x = 3 1 2 3 4 (1) y²µ§|XêÝ  2; (2) ¦Ñëê a§b§c Š§¿¦)§|" 1. N ´ w Ñ §  § | 13!4‡  §  ™  ê X ê Ø ¤ ' ~ § ¤±XêÝ   A Œ u ½  u2¶ Ó ž d uα3 − α1 = (−1, −1, −3, 2) t §α2 − α1 = (3, −2, −1, 5)t ´ ƒ A à g  § | )§ Ø¤'~§¤±àg§|Ä:)X¥–kü ‡‚5Ã')§u´XêÝ u½u4 − 2 = 2§ U´2¶  2. dXêÝ   2 −9 −c b 2 −9 −c        1 −11 a  1 −11 a b   = 2§ 2§¤±r      5 −29 −2 −15   5 −29 −2    4 −31 −1 −15 4 −31 −1 1 5  (
8 )     1 −11 a b 1 −11 a b            0 13  0 13 −2a − c −2a − c −b −b  ⇒       0 26 −5a − 2 −5b − 15   0 0 2c − a − 2 −3b − 15      0 0 c − 2a − 1 −3b − 15 0 13 −4a − 1 −4b − 15 §¤± 2c − a − 2 = 0 , a = 0, c = 1§K −3b − 15 = 0b = ) c − 2a − 1 = 0     11 10 2 2 −9 −1 −5 1  1 0 − 13 − 13 13     5 1  1      1 −11 0 −5 1   0 1 − 13 13 − 13    −5§džO2Ý   ⇒    5 −29 −2 −15 3  0 0 0 0 0       4 −31 −1 −15 3 0 0 0 0 0       2 11 10  13   13   13   1   1   5  C −    −   13   13   13  ¾ §|)  + x3   + x4         ‚  0   1   0  S       Ø 0 0 1 ( ‡ ‰ K 1 6  (
8 ) Ê !(  K10© ©) ® A ´ m × n Ý §P A =  α1 α2 · · · αn  ,Ù ¥ α1 , · · · , αn ´ A •þ|§ T ´?¿ m šÛɐ ¿P T A =   β1 β2 · · · βn " {i1 , · · · , ik } ⊂ {1, · · · , n} ´?¿I8§y² αi1 ,· · · , αik ‚5ƒ' = βi1 ,· · · , βik ‚5ƒ'" 8! ( K10© ©) XJ f (x) = an xn + · · · + a1 x + a0 ∈ K [x]§  §KP f (A) = an An + · · · + a1 A + a0 In §Ù¥ In ü AKþn "yõ‘ª f (x) Ú g (x)  K þ ü‡pƒõ‘ª"y²µàg‚5§|f (A) g (A) X = 0 )˜m V ´àg‚5§|f (A) X = 0 )˜m V1 † g (A) X = 0 )˜m V2 †Ú, =y V = V1 ⊕ V2 " 1 7  (
8 ) Ô !(  K10© ©) XJ A ´ m × n Ý §β1 , β2 ´ü‡‰½•þ" y²§| AX = β1 † AX = β2 Ӟk) " ) e = (A β1 β2 ) ´ m × (n + 2) ©¬Ý A e Ù¥ = R(A) = R(A), ( C ¾ ‚ S Ø ‡ ‰ K 1 8  (
8 ) l ! ( K 10© ©) 阇 ²µ?˜ A§XJ A2 = A§ K¡ A ˜‡˜Ý þŒL«˜‡šÛÉÝ †˜‡˜Ý 1 9  (
8 ) ¦È" "y