行列代数2010A
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1 a ij i j 1) i +j i, j) ij ij 1 j a i1 a ij a i a 1 a j a ij 1) i +j 1,j 1,j +1 a i1,1 a i1,j 1 a i1,j +1 a i1, a i +1,1 a i +1.j 1 a i +1,j +1 a i +1, a 1 a,j 1 a,j +1 a,
2 ij i j 1,j 1,j +1 ij 1) i +j a i1,1 a i1,j 1 a i1,j +1 a i1, a i +1,1 a i +1.j 1 a i +1,j +1 a i +1, a 1 a,j 1 a,j +1 a, 1 j a i1 1 a i a 1 a j a a 1 0 a a i1 i1 + a i2 i2 ++ a i i a i1 j1 + a i2 j 2 ++ a i j 0 for i j a ik ik a ik 1 k k 0 a ik 0 < i a ik ik k1 1 j a i1 a ij a i a 1 a k a a 1 a k a k^ a 1 a j a a ik jk a ik 1 k a 1 a k a 1 k 0 a ik 0 a 1 a k a k^ < j a ik jk k1 1 i j a i1 a ii a ij a i a i1 a ii a ij a i 1 a 1 a i a j a i j ij
3 i 1i + i 2i ++ a i i i 1j + i 2j ++ a i j 0 for i j a ki ki a ki 1 0 a k1 1 a k 1 0 a k1 a ki a k a ki ki k1 1 i a i1 a ii a i a 1 0 a a 1 0 a a 1 a i a a ki kj a ki 1 0 a k1 1 a k a 1 0 a 1 0 a k1 a ki a k a 1 0 a a ki kj k1 1 i j 1 i i a i1 a ii a ii a i a j1 a ji a ji a j 1 a 1 a i a i a i j ij à t E [ a ij ], t [ ij ], k1 a ik jk [ ij ] E [ a ij ], t [ ij ], k1 a ki kj [ ij ] E
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6 cos si 0 si cos x cos, cos 2 + si ) 1+1 cos cos, 12 1) 1+2 si si, 13 1) ) 2 +1 si) si, 22 1) 2 +2 cos cos, 23 1) ) , 32 1) , 33 1) 3 +3 cos 2 + si 2 ) cos si )si cos y si x cos +) cos cos si si x cos y si y si +) cos si + si cos xsi + y cos z z x x y y z z x x y )1 y z z cos si 0 )1 )si cos ) x x a b z a + bi T b a 1 z z a + bi a b a T 1 + T* ) - T, b 1 a bi) a ) T T a b bi + b 2 b, 1 a b a a b + * - 1 a bi) ), a bi a + bi)a bi) 1 a + bi 1 z. T 1 1 z
7 P B O D 0, D 0 P BD 1 O D 1 X X 11 X 12 PX E 2 P 1 X X 21 X 22 PX X 11 + BX 21 X 12 + BX 22 E O DX 21 DX 22 O E X 11 + BX 21 E, X 12 + BX 22 O, DX 21 O, DX 22 E DX 21 O D X 21 O DX 22 E D X 22 D 1 X 11 + BX 21 E, X 21 O X 11 1 X 12 + BX 22 O X 12 1 BX 22 1 BD 1 P BD 1 O D 1 XP E 2 1 O 1, 2,..., 2 p O p X 11 X 12 X 1p 1 O X X 21 X 21 X 2p, 2 X p1 X p2 X pp O p 1 O X 11 X 12 X 1p 1 X 11 1 X 12 1 X 1p X 2 X 21 X 21 X 2p 2 X 21 2 X 22 2 X 2p O p X p1 X p2 X pp p X p1 p X p2 p X pp i i X ii 1 1 X ii i i X ij 0 X ij 1 i O ij O ij i ) j) * 1 O O p 1 1 E E E
8 b 2 + c 2 ab ac ba c 2 + bc ca cb + b 2 a,b,ca,b,c b 2 + c 2 ab ac ba c 2 + bc abc ca cb + b 2 b 2 + c 2 a a a b c 2 + b b c c + b 2 c b 2 + c 2 b 2 c 2 c 2 + c 2 b 2 + b 2 0 b 2 c 2 2c 2 c 2 + c 2 2 2b 2 b 2 + b 2 0 b 2 c 2 c 2 c 2 + c 2 2 b 2 b 2 + b 2 0 b 2 c 2 c 2 c 2 b b 2 2 4a 0 b 2 c 2 c b 2 c 2 b 2 0 x x x x 1 a 4 a 0 a 0 x 4 + x 3 + x 2 + x 1 + a 4 x x x x 1 a 4 a x 2 x x x 1 a 4 + x a 0 x x x 1 a 4 + x a x 3 x x 1 x x 1 a 4 + x + x 2 a 0 a 4 + x + x 2 a x 4 x 1 a 4 + x + x 2 + x 3 a 0 x 4 1 a 4 + x + x 2 + x 3 a 0 a 4 + x + x 2 + x 3 + a 0 x 4
9 O C D D ij P O C D P )* ) 1* 1) 2* 2) * ) + * i* i) 0 for i, p ad *i) > p - i, p. *i), p ) 1,p) ) 1,2,q) / * 0 *1),*p) p + 1) p,p + 2) p,) p ) ) ) P ) 1 1) 2 2) ) ) ) 1 1) 2 2) p p) p+1) p+1) p+ 2) p+ 2) ), ) 1 1) 2 2) p p) ) p+1) p+1) p+ 2) p+ 2) ) D I p I q p i) p 0 i p p i p i i) p 1 1) 2 2) ) 0 B B + B B B B ib + ib B B + B B B + + B B 0 B + B B B B ib B B + i ib B 0 + ib ib + ib
10 , 1 ±1 1 E 1 1 1, 1 ±1 B 1k 1 <k 2 <<k m k1 k m a mk1 a mk m b k1 1 b k1 m b k m 1 b k m m b 11 b 1m B [ a ] b j1 a j b jm a j b 1 b m j 1 j 1 B b j1 a j b jm a j b 11 + b b 1 a b 1m + b 2m ++ b m a j 1 j 1 )* b k1 1 b k 2 2 b k m m a k 1 a k m 1k 1 <k 2 <<k m 1k 1 <k 2 <<k m m k1 k m b k1 1 b k1 m a mk1 a mk m b k m 1 b k m m
11 xx a 3 B a 3 b 11 b 21 b 22 b 31 b 32 b 11 + b 21 + b 31 + b 22 + b 32 a 3 B b 11 + b 21 + b 31 + b 22 + b 32 b 11 b 22 + b 11 b 32 + b 21 + b 21 b 32 + b 31 + b 31 b 22 b 11 b 22 b 21 ) + b 11 b 32 b 31 ) + b 21 b 32 b 31 b 22 ) b 11 b 21 b 22 + b 11 b 31 b 32 + b 21 b 22 b 31 b 32 k 1,k 2 )b b k1 1 k 2 2 a k1 a k2 + k 1,k 2 )b b k1 1 k 2 2 a k1 a k2 + k 1,k 2 )b b k1 1 k 2 2 a k1 a k2 1k 1 <k 2 2 1k 1 <k 2 3 2k 1 <k 2 3 k 1,k 2 )b b k1 1 k 2 2 a k1 a k2 1k 1 <k 2 3 f i x), g i x),h i x) d dx d dx d dx f 1 x) g 1 x) f 2 x) g 2 x) f 1 x) g 1 x) f 2 x) g 2 x) + f 1 x) g 1 x) f 2 x) g 2 x) f 1 x) g 1 x) h 1 x) f 3 x) g 3 x) h 3 x) f 1 x) g 1 x) h 1 x) f 2 x) g 2 x) h 2 x) f 2 x) g 2 x) h 2 x) + f 3 x) g 3 x) h 3 x) f 1 x) g 1 x) f 2 x) g 2 x) d dx f 1x)g 2 x) g 1 x) f 2 x) f 1x) g 1 x) f 2 x) g 2 x) + f 1x) g 1 x) f 2 x) g 2 x) d dx f 1 x) g 1 x) h 1 x) f 2 x) g 2 x) h 2 x) f 3 x) g 3 x) h 3 x) f 1 x) g 1 x) h 1 x) f 2 x) g 2 x) h 2 x) f 3 x) g 3 x) h 3 x) f 1 x) g 1 x) h 1 x) + f 2 x) g 2 x) h 2 x) f 3 x) g 3 x) h 3 x) ) f 1 x)g 2 x) g 1 x) f 2 x)) + f 1 x) g 2 x) g 1 x) f 2 x)) d dx f 1 g 2 h 3 + f 2 g 3 h 1 + f 3 g 1 h 2 f 3 g 2 h 1 f 2 g 1 h 3 f 1 g 3 h 2 ) ) + f 1 g 2 h 3 + f 2 g 3 h 1 + f 3 g 1 h 2 f 3 g 2 h 1 f 2 g 1 h 3 f 1 g 3 h 2 ) + f 1 g 2 h 3 + f 2 g 3 h 1 + f 3 g 1 h 2 f 3 g 2 h 1 f 2 g 1 h 3 f 1 ) f 1 g 2 h 3 + f 2 g 3 h 1 + f 3 g 1 h 2 f 3 g 2 h 1 f 2 g 1 h 3 f 1 g 3 h 2 f 1 x) g 1 x) h 1 x) f 2 x) g 2 x) h 2 x) + f 3 x) g 3 x) h 3 x) f 1 x) g 1 x) h 1 x) f 2 x) g 2 x) h 2 x) f 3 x) g 3 x) h 3 x) g 3 h 2 f 1 x) g 1 x) h 1 x) + f 2 x) g 2 x) h 2 x) f 3 x) g 3 x) h 3 x)
12 1 x x x b 1 1 x x x b 2 a 1 x 1 + a 2 x a x b x b , b a 1 a 2 a b 1 b 2 b
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14 x 1 + x 2 + 2x 3 2 x 1 2x 2 + x 3 0 2x 1 + x 2 x 3 2 x 1 1 x 2 1 x ) ) ) X E X 1 E 1 X [ x 1 x 2 x ], E e 1 e 2 e x 1 x 2 x x k e k k 1,...) [ ] [ ] [ e 1 e 2 e ] e 1 e 2 e [ ] [ E X] X 1 E 1
15 /2 1/2 1/ /3 1/3 2 / /2 3/2 1/ /3 4 /3 2 / /3 4 /3 2 / /3 1/3 2 / /3 1/3 2 / /3 2 /3 2 / /4 1/4 3/ /2 1/2 1/ /4 1/4 1/ /4 3/4 1/ /4 3/4 1/ /4 1/4 3/ /4 1/4 3/4 ) X X
16 /2 1/ /2 1/ /2 0 1/ / / /2 1/ /2 1/ /2 1/ / / ) X /2 1/ /2 0 1/ / / /2 1/ /2 1/ /2 0 1/ / /2
行列代数2010A
(,) A (,) B C = AB a 11 a 1 a 1 b 11 b 1 b 1 c 11 c 1 c a A = 1 a a, B = b 1 b b, C = AB = c 1 c c a 1 a a b 1 b b c 1 c c i j ij a i1 a i a i b 1j b j b j c ij = a ik b kj b 1j b j AB = a i1 a i a ik
弾性定数の対称性について
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(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y
![(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y](/thumbs/98/136512264.jpg "(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y")
(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = a c b d (a c, b d) P = (a, b) O P a p = b P = (a, b) p = a b R 2 { } R 2 x = x, y R y 2 a p =, c q = b d p + a + c q = b + d q p P q a p = c R c b
II 2 3.,, A(B + C) = AB + AC, (A + B)C = AC + BC. 4. m m A, m m B,, m m B, AB = BA, A,, I. 5. m m A, m n B, AB = B, A I E, 4 4 I, J, K
II. () 7 F 7 = { 0,, 2, 3, 4, 5, 6 }., F 7 a, b F 7, a b, F 7,. (a) a, b,,. (b) 7., 4 5 = 20 = 2 7 + 6, 4 5 = 6 F 7., F 7,., 0 a F 7, ab = F 7 b F 7. (2) 7, 6 F 6 = { 0,, 2, 3, 4, 5 },,., F 6., 0 0 a F
(2018 2Q C) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y
![(2018 2Q C) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y](/thumbs/101/151465237.jpg "(2018 2Q C) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y")
(2018 2Q C) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = a c b d (a c, b d) P = (a, b) O P a p = b P = (a, b) p = a b R 2 { } R 2 x = x, y R y 2 a p =, c q = b d p + a + c q = b + d q p P q a p = c R c b
) 9 81
4 4.0 2000 ) 9 81 10 4.1 natural numbers 1, 2, 3, 4, 4.2, 3, 2, 1, 0, 1, 2, 3, integral numbers integers 1, 2, 3,, 3, 2, 1 1 4.3 4.3.1 ( ) m, n m 0 n m 82 rational numbers m 1 ( ) 3 = 3 1 4.3.2 3 5 = 2
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０４年度LS民法Ⅰ教材改訂版.PDF
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LINEAR ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University
LINEAR ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University 2002 2 2 2 2 22 2 3 3 3 3 3 4 4 5 5 6 6 7 7 8 8 9 Cramer 9 0 0 E-mail:hsuzuki@icuacjp 0 3x + y + 2z 4 x + y
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新たな基礎年金制度の構築に向けて
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yoshi@image.med.osaka-u.ac.jp http://www.image.med.osaka-u.ac.jp/member/yoshi/ II Excel, Mathematica Mathematica Osaka Electro-Communication University (2007 Apr) 09849-31503-64015-30704-18799-390 http://www.image.med.osaka-u.ac.jp/member/yoshi/
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熊本県数学問題正解
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(Basic of Proability Theory). (Probability Spacees ad Radom Variables , (Expectatios, Meas) (Weak Law
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