(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
|
|
- そう おうじ
- 4 years ago
- Views:
Transcription
1 (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 c p = ca cb p c a p = c q = b d p q = ac + bd p p = a 2 + b 2 0 p p p p p q θ p q = p q cos θ 1
2 2 (2018 2Q C) a, b (a, b) a a f : R 2 R 2 x y = f(x) g : R 2 R 2 f + g : R 2 R 2 (f + g)(x) = f(x) + g(x) f g k R kf : R 2 R 2 (kf)(x) = kf(x) f k R 2 ( ) X = (x 1, x 2 ) Y = (y 1, y 2 ) F : R 2 R 2 { y1 = ax 1 + bx 2 + e 1 y 2 = cx 1 + dx 2 + e 2 a, b, c, d, e 1, e 2 x1 (e 1, e 2 ) X x = (e x 1, e 2 ) Y ( 2 ) ax1 + bx 2 cx 1 + dx 2 f : R 2 R 2 x1 y1 x = y = = f(x) x 2 y 2 { y1 = ax 1 + bx 2 y 2 = cx 1 + dx 2 a b x 1, x 2 f A = c d s t g : R 2 R 2 x1 B = x = u v x 2 ax1 + bx (f + g)(x) = f(x) + g(x) = 2 sx1 + tx + 2 cx 1 + dx 2 ux 1 + vx 2 (a + s)x1 + (b + t)x = 2 (c + u)x 1 + (d + v)x 2 f + g A B a + s b + t f A g B c + u d + v a + s b + t A + B = c + u d + v
3 (2018 2Q C) 3 k kf ax1 + bx (kf)(x) = kf(x) = k 2 kax1 + kbx = 2 cx 1 + dx 2 kcx 1 + kdx 2 ka kb kf f A A k kc kd ka kb ka = kc kd f : R 2 R 2, g : R 2 R 2 1 f, g a b s t A =, B = c d u v f g g f x1 x = x 2 (g f)(x) = g(f(x)) ax1 + bx f(x) = 2 cx 1 + dx 2 s(ax1 + bx (g f)(x) = g(f(x)) = 2 ) + t(cx 1 + dx 2 ) u(ax 1 + bx 2 ) + v(cx 1 + dx 2 ) (sa + tc)x1 + (sb + td)x = 2 (ua + vc)x 1 + (ub + vd)x 2 g f sa + tc sb + td ua + vc ub + vd g B f A sa + tc sb + td B A = ua + vc ub + vd B A BA s t a b sa + tc sb + td BA = = u v c d ua + vc ub + vd B A BA
4 4 (2018 2Q C) a b A = c d A (a b) ( A ) 1 (c d) A 2 a b A A 1 A 2 A c d i j ( A ) (i, ( j) ) a b s t 2 A =, B = A B c d u v a + s b + t A + B = c + u d + v k A k ka kb ka = = Ak kc kd B A sa + tc sb + td BA = ua + vc ub + vd BA (i, j) B i ( ) A j 1 A f x1 x = x 2 ax1 + bx f(x) = 2 a b x1 = cx 1 + dx 2 c d x 2 f(x) = Ax 0 0 O 0 0 A, B, C (AB)C = A(BC) (A + B)C = AC + BC, A(B + C) = AB + AC 1 0 E 0 1 AB = BA A O B O AB = O
5 [ ] R n n x = (2018 2Q C) 5 R m y = f(x) y 1 = a 11 x 1 + a 12 x a 1n x n y 2 = a 21 x 1 + a 22 x a 2n x n y m = a m1 x 1 + a m2 x a mn x n x 1,, x n x 1 x n a 11 a 12 a 1n a A = 21 a 22 a 2n a m1 a m2 a mn R n A m n A f y = f(x) = a 11 x 1 + a 12 x a 1n x n a 21 x 1 + a 22 x a 2n x n a m1 x 1 + a m2 x a mn x n f A f(x) = A(x) = Ax a 11 a 12 a 1n a y = Ax = 21 a 22 a 2n a m1 a m2 a mn A x A x f : R n R m g : R n R m A, B (f + g)(x) = f(x) + g(y) A + B A B A B m A B n h: R m R l C (h f)(x) = h(f(x)) CA C A m g f : R n R l CA l n x 1 x n
6 6 (2018 2Q C) mn () {a ij 1 i m, 1 j n} m n a 11 a 12 a 1n a A = 21 a 22 a 2n a m1 a m2 a mn m n m n (m, n) m n i A i j A j A i j a ij A (i, j) (i, j) a ij (a ij ) m = n n n 2 A = (a ij ), B = (b ij ) i, j a ij = b ij A B A = B A, B A, B m n A = (a ij ), B = (b ij ) A + B = (a ij + b ij ) m n A = (a ij ) c A c ca = (ca ij ) c A, B A B AB A = (a ik ) l m B = (b kj ) m n AB (i, j) c ij c ij = a i1 b 1j + a i2 b 2j + + a im b mj a 11 a 1m i a i1 a ik a im a l1 a lm j a 11 a 1j a 1n a kj = (c ij ) a m1 a mj a mn 0 O m n O m,n 0 o (i, i) E n E n
7 (2018 2Q C) 7 [ ] A = (a ij ) m n a 11 a 1j a 1n A = a i1 a ij a in a m1 a mj a mn A (i, j) (j, i) n m A t A A A a 11 a i1 a m1 a 11 a 1j a 1n t A = a 1j a ij a mj, A = a i1 a ij a in a 1n a in a mn a m1 a mj a mn ( a + bi = a bi, i = 1) t A A A A = t A ( ) A, B (1) t (AB) = t B t A (2) (AB) = B A ( ) A (1) A t A = A (2) A t A = A (3) A A = A (4) A A = A A = (a ij ) i = j a ii A 0 ( ) A A 0 a 1 O A = O a n A = (a ij ) i > j a ij = 0 A i < j a ij = 0 A a 11 a 1n O a nn, a 11 O a n1 a nn
8 8 (2018 2Q C) [ ] ( ) n A AX = E XA = E n X X A A 1 () a 0 ax = b x b a n A AX = B X X = EX = (A 1 A)X = A 1 B Y A = B Y Y = Y E = Y (AA 1 ) = BA X Y A =, B = A 1 = AX = B X X = A B = Y A = B Y Y = BA = 1 1 A AX = B XA = B X X n A R n R n f(x) = Ax n E n R n R n id R n A AX = XA = E n X X g f g = g f = id R n R n R n g f g f ( ) A, B (1) A 1 (A 1 ) 1 = A (2) AB (AB) 1 = B 1 A 1 (3) A (A ) 1 = (A 1 )
9 (2018 2Q C) 9 [] m n A A = b 1 b 2 {}}{{}}{ b k {}}{ a 1 { A 11 A 12 A 1k a 2 { A 21 A 22 A 2k a l { A l1 A l2 A lk i A i1,, A ik a i j A 1j,, A lj b j a i = m, b j = n m n B A A B A 11 + B 11 A 12 + B 12 A 1k + B 1k A A + B = 21 + B 21 A 22 + B 22 A 2k + B 2k A l1 + B l1 A l2 + B l2 A lk + B lk n s B A B = b 1 { B 11 B 12 B 1p b 2 { B 21 B 22 B 2p b k { B k1 B k2 B kp j A j A B AB (i, j) A i1 B 1j + A i2 B 2j + + A ik B kj n i a i = b i a 1 a 2 a k {}}{{}}{{}}{ a 1 { A 11 A 12 A 1k a A = 2 { A 21 A 22 A 2k a k { A k1 A k2 A kk
10 10 (2018 2Q C) A = (a ij ) m n A i A a (i) = ( a i1 a i2 a in ) A = a (1) a (2) a (m) A n l B = (b jk ) k b 1k b b k = 2k b nk B B = ( b 1 b 2 b l ) B a (1) (1) ( x 1 x m ) A = ( x 1 x m ) = x 1 a (1) + + x m a (m) a (m) x 1 x 1 (2) B = ( b 1 b l ) = x 1 b x l b l x l (3) AB = A ( b 1 b l ) = ( Ab 1 Ab l ) a (1) (4) AB = B = a (m) a (1) B a (m) B x l a (1) a (1) b 1 a (1) b l (5) AB = ( b 1 b l ) = a (m) a (m) b 1 a (m) b l j 1 0 e j e 1 =, e 2 = 0, 0 0 n E n E n = ( e 1 e 2 e n )
11 (2018 2Q C) 11 [ ] E n n E ij (i, j) 1 0 n ( ) P ij = E n E ii E jj + E ij + E ji = 1 O O 1 1 O Q i (c) = E n + (c 1)E ii = c (c 0), O 1 1 O 1 c R ij (c) = E n + ce ij = (i j) 1 O 1, 1 P ij P ij = E n, R ij (c)r ij ( c) = R ij ( c)r ij (c) = E n c 0 Q i (c)q i (1/c) = Q i (1/c)Q i (c) = E n A P B = P A P 1 A = P 1 B A P A = B
12 12 (2018 2Q C) a (1) A n m A = ( a (i) ) a (n) P ij A = i j a (j) a (i), Q i (c)a = i ca (i), R ij (c)a = (1) i j (r i r j ) (2) i c 0 (cr i ) (3) i j c (r i + cr j ) i a (i) + ca (j) j a (j) a (i) a (j) r i r j a (j) a (i), a (i) cr i ca (i), a (i) a (j) r i +cr j a (i) + ca (j) a (j) A (1) i j (c i c j ) (2) i d 0 (dc i ) (3) j i c (c j + dc i ) (1) (2) 0 (3) 1 (3)
13 [ ] A (1) i j (2) i c 0 (3) i j c (2018 2Q C) 13 ( ) A = (A ij ) A ij m n A mi + 1 (m + 1)i c 0 A i c k = 1,, m A mi + k mj + k A i j A i1 A it X ca i1 ca it, A i1 A j1 A it A jt A j1 A i1 A jt A it R = i j O i E m C, Q = j E m O i O ( i X ) O A A i1 A j1 A i1 A it A jt A it R Q A i1 + CA j1 A it + CA jt A j1 A jt XA i1 XA it P A A 1i A 1j A 1i A 1j + A 1i C A sj + A si C A si A sj C A si,
14 14 (2018 2Q C) [ ] A m n A A t o 1 A 11 0 A A 1r 0 t o 1 A A 2r (141) 0 t o 1 0 A 3r 0 1 A rr O O O A ij t o A ij 0 A = (a ij ) m n A o j A A = (O a j a n ) a j o a ij 0 i 1/a ij (3) a 1j = = a i 1 j = a i+1 j = = a mj = 0 ( j( (i, j) ) i 1 t ) o 1 A A O o A 1 1 t o 1 A 11 A 12 A 1r 0 t o 1 A 22 A 2r 0 t o 1 A 3r 0 1 A rr O O O o (141) ( ) P 1,, P k P k P 2 P 1 A P = P k P 2 P 1 A P P A A B A B B A A (141) A r A rank A
15 [ ] x 1, x n (151) (2018 2Q C) 15 a 11 x 1 + a 12 x a 1n x n = c 1 a 21 x 1 + a 22 x a 2n x n = c 2 a m1 x 1 + a m2 x a mn x n = c m m n A = (a ij ), n x = (x i ) m c = (c i ) Ax = c A (151) A c (A c) (151) Ax = c x A A 1 A 1 x = (A 1 A)x = A 1 c A 1 (A c) A 1 A (A c) A Ax = c x (A c) (152) (A c) (A d) = 1 A 11 0 A A 1r d 1 0 O 1 A A 2r d 2 0 O 1 0 A 3r d A rr d r O O O d () P Ax = c P Ax = P c A x = d A x = d x Ax = c x A x = d A x = d x Ax = c d o Ax = c d = o Ax = c A x = d
16 16 (2018 2Q C) d = o (A d) i 1 = 1 < i 2 < < i r x i1 x 1 x = x i2 x ir x r A x = d x i1 = d 1 A 11 x 1 A 12 x 2 A 1r x r x i2 = d 2 A 22 x 2 A 2r x r x ir = d r A rr x r A = (a ij ) x = x i1 x 1 x i2 x 2 x ir x r d 1 o d 2 = ọ d r o d 1 A 11 x 1 A 12 x 2 A 1r x r x 1 d 2 A 22 x 2 A 2r x r = x 2 d r A rr x r + j i 1,,i r x j a 1j o a 2j ọ a rj o + e j, (x j ) x j n r x x r x j c j x = x 0 + c 1 a c n r a n r, (c j )
17 (2018 2Q C) 17 Ax = o x = o Ax = o Ax = o s c 1,, c s x = c 1 a c s a s Ax = o a 1,, a s Ax = o Ax = o V = {x Ax = o} = {c 1 a c s a s c 1,, c s } Ax = o Ax = c x = d d Ax = c A(x d) = o x d Ay = o x = d + y Ax = c () + ( Ax = o ) 1 0 a d b d x 1 + ax 3 = d 1 x 2 + bx 3 = d 2 0 = 0 x 1 = d 1 ax 3 x 2 = d 2 bx 3 x 3 = x 3 c = x 3 x 1 x 2 = d 1 d 2 + c a b, (c ) x (152) ( ) x n Ax = c (1) rank(a c) = rank A = n (2) rank(a c) = rank A < n n rank A (3) rank(a c) = rank A + 1
18 18 (2018 2Q C) [ ] AX = B (A B) AX = B X A, B (A B) (A B ) = O 1 A 11 0 A A 1r b (1) 0 O 1 A A 2r b (2) 0 O 1 0 A 3r b (3) 0 1 A rr b (r) O O O B 1 (b (i) B 1 ) B 1 = O AX = B AX = B A X = B A X = B X AX = B X, B X = (x 1 x n ), B = (b 1 b n) A x j = b j x j = y j + s c kj a k k=1 X = (x 1 x n ) = (y 1 y n ) + (a 1 a s )(c kj ) k,j Y = (y 1 y n ) AX = B 1 a i Aa i = o E ij (a 1 a s )E ij AX = B Ax = o s AX = B ns XA = B t A t X = t B ( t A t B) t X XA = B X
19 (2018 2Q C) 19 [ ] A n A E n P 1,, P k P k P 1 A = E n P 1 k A = P1 1 P 1 k,, P 1 1 E n = P 1 1 P 1 k AP k P 1 = (P1 1 P 1 )(P k P 1 ) = E n P = P k P 1 P A = E n AP = E n A A 1 = P A A E n A P k P k 1 P 2 P 1 E n = P k P k 1 P 2 P 1 = P = A 1 A A 1 (A E n ) A E n P k P k 1 P 2 P 1 (A E n ) = P k P k 1 P 2 (P 1 A P 1 ) = (P k P k 1 P 2 P 1 A P k P k 1 P 2 P 1 ) = (P A P ) = (E n A 1 ) A A 1 E n n A E n A n 1 A n A 0 (A E n ) 0 P P A = P A P O A A E n (A E n ) (E n X) X = A 1 (A E n ) 0 A k
20 20 (2018 2Q C) ( ) A n E n n n 2n (A E n ) (1) (A E n ) (E n X) A A 1 X (2) (A E n ) (A X) A 0 A n 1 A 1 1/c (2) 1 a, b x, y ax + by = 1 a, b d x, y ax + by = d A = a 11 ai1 = 1 a n1 a 11 1 r 1 r i 1 a 11 r j a j1 r 1 (j 2) (a i1 =a 11,a j1 =a j1,j i) a i1, a j1 x, y xa i1 + ya j1 = 1 x = y = 0 y 0 (x 0 ) a i1 a j1 yr j a i1 ya j1 r j +xr i 1 a 11,, a n1 d 1 d a i1
21 (2018 2Q C) 21 [ ] A 0 {}}{ 0 {}}{ 0 {}}{ 0 {}}{ O 1 A 11 0 A A 1r 0 O 1 A A 2r 0 O 1 0 A 3r 0 1 A rr O O O A rank A ( ) 1 2 r 1 O 1 1 = Er O O O O O A A A r A ( ) Er O A A O O r A m n A P, Q P, Q Er O P AQ = O O r = rank A P, Q P AQ Er O O O Er O P, Q P AQ = rank A = r O O
22 22 (2018 2Q C) A n rank A = n A n A A n A n A En = A O n A = E n n n P = P k P 1 P A = E n A A, X n AX = E n P P A = B P = P E n = P AX = BX P P B B n B = E n P = X XA = P A = B = E n AX = E n XA = E n A X = A 1 X 1 = A XA = E n X = A 1 A n AX = E n XA = E n X A X = A 1 ( ) n A (1) A (2) A E n (3) rank A = n (4) XA = E n AX = E n n X (4) n X A A = P 1 1 P 1 k P 1 k,, P 1 1 ( ) A m n B n l P 1,, P k (1) 0 rank A min{m, n} rank A = 0 A = O (2) rank( t A) = rank A, rank(a) = rank A (3) P, Q rank(p A) = rank(aq) = rank(p AQ) = rank A (4) rank(ab) min{rank A, rank B}
23 (2018 2Q C) 23 [ ] n {1, 2,, n} S n S n σ S n p i {1,, n} i j p i p j σ = ( p 1 p 2 p i p n ) σ i p i σ(i) σ S n i < j σ(i) > σ(j) σ(i) σ(j) {σ(i), σ(j)} t(σ) t(σ) σ = ( p 1 p 2 p i p n ) t(σ) = n (p i p i ) i=1 σ S n σ sgn(σ) sgn(σ) = ( 1) t(σ) n A = (a ij ) A A det A A = σ S n sgn(σ)a 1σ(1) a 2σ(2) a nσ(n) σ S n n! S n σ n = 2 {1, 2} σ 1 = (1 2) σ 2 = (2 1) 0, 1 sgn(σ 1 ) = 1, sgn(σ 2 ) = 1 n = 3 A = sgn(σ 1 )a 11 a 12 + sgn(σ 2 )a 12 a 21 = a 11 a 22 a 12 a 21 S 3 = {(1 2 3), (2 3 1), (3 1 2), (1 3 2), (2 1 3), (3 2 1)} 1 1 A = a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 a 11 a 23 a 32 a 12 a 21 a 33 a 13 a 22 a 31 n = 2, 3 () n 4
24 24 (2018 2Q C) [ ] (0 ) (I) 0 0 a 11 a 12 a 1n a n1 a n2 a nn = a 11 0 a 1n a 21 0 a 2n a n1 0 a nn = 0 (II) 1 1 (1, 1) 0 a 11 a 12 a 1n 0 a 22 a 2n 0 a n2 a nn = a a 21 a 22 a 2n a n1 a n2 a nn = a 11 a 22 a 2n a n2 a nn (1) a 1i a 1j a ni a nj = a 1j a 1i a nj a ni (2) 1 c c 1 a 11 ca 1i a 1n a n1 ca ni a nn = c a 11 a 1i a 1n a n1 a ni a nn (3) c a 1i a 1j a ni a nj = a 1i a 1j + ca 1i a ni a nj + ca ni i j j i j 0 0 (I) (II)
25 [] (2018 2Q C) 25 (1) t A = A (2) A = A (3) AB = A B (4) A + B = A + B ( ) a 1 a i 1 a i + a i a i+1 a n = a 1 a i a n + a 1 a i a n a j A B (5) X X = C D X = AD CB, X = A D B C A, B, C, D n c (1) A B O D = A D A C O ( t ) ( D = t A t C t ) O t = A t C D O t = t A t D = A D D (2) 1 i n A B C D i n + i 1 A B C D = C D ( 1)n A B (3) 1 i n A B C D i c c c ca cb C D = A B cn C D (4) 1 i n A B C D i c n + i A B C D = A B C + ca D + cb (5) X n (1) E n X O = 1 A C A C B D = E n O B D = A C X E n A C B D E n O E n B D = A + XC B + XD C D X = A AX + B C CX + D E n X
26 26 (2018 2Q C) [ ] n A A A 1 AA 1 = E AA 1 = A A 1 = E = 1 A = 0 A rank A < n P P A = P A = 0 P P = 0 A = 0 A A = 0 ( ) n A (1) A (2) AX = XA = E n X (3) AX = E n XA = E n X (4) A E n (5) A E n (6) A E n (7) rank A = n (8) Ax = o (9) Ax = c (10) A = 0 n A = (a ij ) A tr A tr A = n a ii = a 11 + a a nn i=1 A = (a ij ), B = (b ij ) n AB (i, i) n n n n tr(ab) = a ij b ji = b ji a ij = tr(ba) i=1 j=1 j=1 i=1 n a ij b ji j=1 ( ) A, B, C n P n c (1) tr (A + B) = tr A + tr B (2) tr (ca) = c tr A (3) tr (AB) = tr (BA) (4) tr (ABC) = tr (BCA) (5) tr (P 1 AP ) = tr (A)
27 (2018 2Q C) 27 [ ] 1 1 (1, 1) 0 a 11 a 12 a 1n a a 0 a 22 a 2n a = 21 a 22 a 2n 22 a 2n = a 11 a 0 a n2 a nn a n1 a n2 a n2 a nn nn i j 0 A 11 o A 12 a ij A 21 o A 22 = A 11 A 12 t o a t ij o A 21 A 22 = ( 1)i+j a ij A 11 A 12 A 22 A 22 A = (a ij ) n A a ij ã ij A i j n 1 ( 1) i+j ( ) n A = (a ij ) (i, j) ã ij a 11 a 1 j 1 a 1 j+1 a 1n ã ij = ( 1) i+j a i 1 1 a i 1 j 1 a i 1 j+1 a i 1 n a i+1 1 a i+1 j 1 a i+1 j+1 a i+1 n a n1 a n j 1 a n j+1 a nn A = (a ij ) i a ij 0 A = a ij ã ij A = (a ij ) j a ij 0 A = a ij ã ij
28 28 (2018 2Q C) A 1 a + b A 2 = A 1 a A 2 + A 1 b A 2 A = (a ij ) i a (i) = a i1 t e a in t e n A = a i1 ã i1 + + a in ã in ( ) A = (a ij ) (i, j) ã ij (i, j) A = A = n a ik ã ik = k=1 n a kj ã kj k=1 n a ik ã ik i A = k=1 j 1 1 A = a 11 ã 11 + a 12 ã a 1n ã 1n n a kj ã kj k=1 () = a 11 ã 11 + a 21 ã a n1 ã n1 () n A (i, j) A (j, i) A ( ) A Ã ( ) ã 11 ã n1 Ã = ã 1n ã nn AÃ = ÃA = A E n A = 0 A A 1 = 1 A Ã
29 [ ] n n n A = (a 1 a 2 a n ), (2018 2Q C) 29 a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2 a n1 x 1 + a n2 x a nn x n = b n a i = a 1i a 2i a ni b = b 1 b 2 b n, x = Ax = b A ( ) A = (a 1 a 2 a n ) n Ax = b x i = a 1 a i 1 b a i+1 a n a 1 a i 1 a i a i+1 a n n m ( m n) a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2 a m1 x 1 + a m2 x a mn x n = b m x i n m a 1i1 x i1 + a 1i2 x i2 + + a 1im x im = b 1 + a 1im+1 x im a 1in x in a 2i1 x i1 + a 2i2 x i2 + + a 2im x im = b 2 + a 2im+1 x im a 2in x in a mi1 x i1 + a mi2 x i2 + + a mim x im = b m + a mim+1 x im a min x in ( det(ajij ) 0 ) x 1 x 2 x n
30 30 (2018 2Q C) [] x, y 2 x, y A = (x y) x, y 2 S S = det A x y det A > 0 x y det A < 0 x, y 2 det A x, y, z 3 x, y, z A = (x y z) 3 x = (x, y) x y x 1 x 2 x 3, y = y 1 y 2 y 3 (x, y) = x 1 y 1 + x 2 y 2 + x 3 y 3, x y = x 2y 3 x 3 y 2 x 3 y 1 x 1 y 3 x 1 y 2 x 2 y 1 (x, x y) = x 1 (x 2 y 3 x 3 y 2 ) + x 2 (x 3 y 1 x 1 y 3 ) + x 3 (x 1 y 2 x 2 y 1 ) x = x 2 y 2 1 x 3 y 3 x 2 x 1 y 1 x 3 y 3 + x 3 x 1 y 1 x 2 y 2 = x 1 x 1 y 1 x 2 x 2 y 2 x 3 x 3 y 3 = 0 (y, x y) = 0 x y x, y x y z x y z = (x, y z) x, y, z () det A (x, y, z) det A > 0, det A < 0 2 A = (a 1, a 2 ), B = (b 1, b 2 ) A, B (a 2 b 2 )(x b 1 ) (a 1 b 1 )(y b 2 ) = 0 (a 2 b 2 )(x b 1 ) (a 1 b 1 )(y b 2 ) a 1 b 1 x = (a 2 b 2 )x (a 1 b 1 )y + (a 1 b 2 a 2 b 1 ) = a 2 b 2 y A, B a 1 b 1 x a 2 b 2 y = 0
(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
More informationlinearal1.dvi
19 4 30 I 1 1 11 1 12 2 13 3 131 3 132 4 133 5 134 6 14 7 2 9 21 9 211 9 212 10 213 13 214 14 22 15 221 15 222 16 223 17 224 20 3 21 31 21 32 21 33 22 34 23 341 23 342 24 343 27 344 29 35 31 351 31 352
More informationII 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
More information2 (2016 3Q N) c = o (11) Ax = b A x = c A n I n n n 2n (A I n ) (I n X) A A X A n A A A (1) (2) c 0 c (3) c A A i j n 1 ( 1) i+j A (i, j) A (i, j) ã i
[ ] (2016 3Q N) a 11 a 1n m n A A = a m1 a mn A a 1 A A = a n (1) A (a i a j, i j ) (2) A (a i ca i, c 0, i ) (3) A (a i a i + ca j, j i, i ) A 1 A 11 0 A 12 0 0 A 1k 0 1 A 22 0 0 A 2k 0 1 0 A 3k 1 A rk
More informationi I II I II II IC IIC I II ii 5 8 5 3 7 8 iii I 3........................... 5......................... 7........................... 4........................ 8.3......................... 33.4...................
More information行列代数2010A
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, ij i j 1,j 1,j +1 ij 1) i +j a
More informationLINEAR 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
More informationA S- hara/lectures/lectures-j.html r A = A 5 : 5 = max{ A, } A A A A B A, B A A A %
A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r A S- 3.4.5. 9 phone: 9-8-444, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office
More informationver Web
ver201723 Web 1 4 11 4 12 5 13 7 2 9 21 9 22 10 23 10 24 11 3 13 31 n 13 32 15 33 21 34 25 35 (1) 27 4 30 41 30 42 32 43 36 44 (2) 38 45 45 46 45 5 46 51 46 52 48 53 49 54 51 55 54 56 58 57 (3) 61 2 3
More information数学Ⅱ演習(足助・09夏)
II I 9/4/4 9/4/2 z C z z z z, z 2 z, w C zw z w 3 z, w C z + w z + w 4 t R t C t t t t t z z z 2 z C re z z + z z z, im z 2 2 3 z C e z + z + 2 z2 + 3! z3 + z!, I 4 x R e x cos x + sin x 2 z, w C e z+w
More informationさくらの個別指導 ( さくら教育研究所 ) A 2 P Q 3 R S T R S T P Q ( ) ( ) m n m n m n n n
1 1.1 1.1.1 A 2 P Q 3 R S T R S T P 80 50 60 Q 90 40 70 80 50 60 90 40 70 8 5 6 1 1 2 9 4 7 2 1 2 3 1 2 m n m n m n n n n 1.1 8 5 6 9 4 7 2 6 0 8 2 3 2 2 2 1 2 1 1.1 2 4 7 1 1 3 7 5 2 3 5 0 3 4 1 6 9 1
More informationad bc A A A = ad bc ( d ) b c a n A n A n A A det A A ( ) a b A = c d det A = ad bc σ {,,,, n} {,,, } {,,, } {,,, } ( ) σ = σ() = σ() = n sign σ sign(
I n n A AX = I, YA = I () n XY A () X = IX = (YA)X = Y(AX) = YI = Y X Y () XY A A AB AB BA (AB)(B A ) = A(BB )A = AA = I (BA)(A B ) = B(AA )B = BB = I (AB) = B A (BA) = A B A B A = B = 5 5 A B AB BA A
More informationJanuary 27, 2015
e-mail : kigami@i.kyoto-u.ac.jp January 27, 205 Contents 2........................ 2.2....................... 3.3....................... 6.4......................... 2 6 2........................... 6
More information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
More information20 9 19 1 3 11 1 3 111 3 112 1 4 12 6 121 6 122 7 13 7 131 8 132 10 133 10 134 12 14 13 141 13 142 13 143 15 144 16 145 17 15 19 151 1 19 152 20 2 21 21 21 211 21 212 1 23 213 1 23 214 25 215 31 22 33
More informationIMO 1 n, 21n n (x + 2x 1) + (x 2x 1) = A, x, (a) A = 2, (b) A = 1, (c) A = 2?, 3 a, b, c cos x a cos 2 x + b cos x + c = 0 cos 2x a
1 40 (1959 1999 ) (IMO) 41 (2000 ) WEB 1 1959 1 IMO 1 n, 21n + 4 13n + 3 2 (x + 2x 1) + (x 2x 1) = A, x, (a) A = 2, (b) A = 1, (c) A = 2?, 3 a, b, c cos x a cos 2 x + b cos x + c = 0 cos 2x a = 4, b =
More informationall.dvi
5,, Euclid.,..,... Euclid,.,.,, e i (i =,, ). 6 x a x e e e x.:,,. a,,. a a = a e + a e + a e = {e, e, e } a (.) = a i e i = a i e i (.) i= {a,a,a } T ( T ),.,,,,. (.),.,...,,. a 0 0 a = a 0 + a + a 0
More informationn ( (
1 2 27 6 1 1 m-mat@mathscihiroshima-uacjp 2 http://wwwmathscihiroshima-uacjp/~m-mat/teach/teachhtml 2 1 3 11 3 111 3 112 4 113 n 4 114 5 115 5 12 7 121 7 122 9 123 11 124 11 125 12 126 2 2 13 127 15 128
More informationuntitled
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/
More information+ 1 ( ) I IA i i i 1 n m a 11 a 1j a 1m A = a i1 a ij a im a n1 a nj a nm.....
+ http://krishnathphysaitama-uacjp/joe/matrix/matrixpdf 1 ( ) I IA i i i 1 n m a 11 a 1j a 1m A = a i1 a ij a im a n1 a nj a nm (1) n m () (n, m) ( ) n m B = ( ) 3 2 4 1 (2) 2 2 ( ) (2, 2) ( ) C = ( 46
More information熊本県数学問題正解
00 y O x Typed by L A TEX ε ( ) (00 ) 5 4 4 ( ) http://www.ocn.ne.jp/ oboetene/plan/. ( ) (009 ) ( ).. http://www.ocn.ne.jp/ oboetene/plan/eng.html 8 i i..................................... ( )0... (
More information04年度LS民法Ⅰ教材改訂版.PDF
?? A AB A B C AB A B A B A B A A B A 98 A B A B A B A B B A A B AB AB A B A BB A B A B A B A B A B A AB A B B A B AB A A C AB A C A A B A B B A B A B B A B A B B A B A B A B A B A B A B A B
More information1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C
0 9 (1990 1999 ) 10 (2000 ) 1900 1994 1995 1999 2 SAT ACT 1 1990 IMO 1990/1/15 1:00-4:00 1 N 1990 9 N N 1, N 1 N 2, N 2 N 3 N 3 2 x 2 + 25x + 52 = 3 x 2 + 25x + 80 3 2, 3 0 4 A, B, C 3,, A B, C 2,,,, 7,
More informationPart () () Γ Part ,
Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35
More informationII A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
More information1. 2 P 2 (x, y) 2 x y (0, 0) R 2 = {(x, y) x, y R} x, y R P = (x, y) O = (0, 0) OP ( ) OP x x, y y ( ) x v = y ( ) x 2 1 v = P = (x, y) y ( x y ) 2 (x
. P (, (0, 0 R {(,, R}, R P (, O (0, 0 OP OP, v v P (, ( (, (, { R, R} v (, (, (,, z 3 w z R 3,, z R z n R n.,..., n R n n w, t w ( z z Ke Words:. A P 3 0 B P 0 a. A P b B P 3. A π/90 B a + b c π/ 3. +
More informationv v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i
1. 1 1.1 1.1.1 1.1.1.1 v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) R ij R ik = δ jk (4) δ ij Kronecker δ ij = { 1 (i = j) 0 (i j) (5) 1 1.1. v1.1 2011/04/10 1. 1 2 v i = R ij v j (6) [
More information20 6 4 1 4 1.1 1.................................... 4 1.1.1.................................... 4 1.1.2 1................................ 5 1.2................................... 7 1.2.1....................................
More informationuntitled
0. =. =. (999). 3(983). (980). (985). (966). 3. := :=. A A. A A. := := 4 5 A B A B A B. A = B A B A B B A. A B A B, A B, B. AP { A, P } = { : A, P } = { A P }. A = {0, }, A, {0, }, {0}, {}, A {0}, {}.
More information2012 A, N, Z, Q, R, C
2012 A, N, Z, Q, R, C 1 2009 9 2 2011 2 3 2012 9 1 2 2 5 3 11 4 16 5 22 6 25 7 29 8 32 1 1 1.1 3 1 1 1 1 1 1? 3 3 3 3 3 3 3 1 1, 1 1 + 1 1 1+1 2 2 1 2+1 3 2 N 1.2 N (i) 2 a b a 1 b a < b a b b a a b (ii)
More information2 7 V 7 {fx fx 3 } 8 P 3 {fx fx 3 } 9 V 9 {fx fx f x 2fx } V {fx fx f x 2fx + } V {{a n } {a n } a n+2 a n+ + a n n } 2 V 2 {{a n } {a n } a n+2 a n+
R 3 R n C n V??,?? k, l K x, y, z K n, i x + y + z x + y + z iv x V, x + x o x V v kx + y kx + ky vi k + lx kx + lx vii klx klx viii x x ii x + y y + x, V iii o K n, x K n, x + o x iv x K n, x + x o x
More information.3. (x, x = (, u = = 4 (, x x = 4 x, x 0 x = 0 x = 4 x.4. ( z + z = 8 z, z 0 (z, z = (0, 8, (,, (8, 0 3 (0, 8, (,, (8, 0 z = z 4 z (g f(x = g(
06 5.. ( y = x x y 5 y 5 = (x y = x + ( y = x + y = x y.. ( Y = C + I = 50 + 0.5Y + 50 r r = 00 0.5Y ( L = M Y r = 00 r = 0.5Y 50 (3 00 0.5Y = 0.5Y 50 Y = 50, r = 5 .3. (x, x = (, u = = 4 (, x x = 4 x,
More information2 2 MATHEMATICS.PDF 200-2-0 3 2 (p n ), ( ) 7 3 4 6 5 20 6 GL 2 (Z) SL 2 (Z) 27 7 29 8 SL 2 (Z) 35 9 2 40 0 2 46 48 2 2 5 3 2 2 58 4 2 6 5 2 65 6 2 67 7 2 69 2 , a 0 + a + a 2 +... b b 2 b 3 () + b n a
More information案内(最終2).indd
1 2 3 4 5 6 7 8 9 Y01a K01a Q01a T01a N01a S01a Y02b - Y04b K02a Q02a T02a N02a S02a Y05b - Y07b K03a Q03a T03a N03a S03a A01r Y10a Y11a K04a K05a Q04a Q05a T04b - T06b T08a N04a N05a S04a S05a Y12b -
More information直交座標系の回転
b T.Koama x l x, Lx i ij j j xi i i i, x L T L L, L ± x L T xax axx, ( a a ) i, j ij i j ij ji λ λ + λ + + λ i i i x L T T T x ( L) L T xax T ( T L T ) A( L) T ( LAL T ) T ( L AL) λ ii L AL Λ λi i axx
More informationx, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y)
x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 1 1977 x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) ( x 2 y + xy 2 x 2 2xy y 2) = 15 (x y) (x + y) (xy
More information2 (1) a = ( 2, 2), b = (1, 2), c = (4, 4) c = l a + k b l, k (2) a = (3, 5) (1) (4, 4) = l( 2, 2) + k(1, 2), (4, 4) = ( 2l + k, 2l 2k) 2l + k = 4, 2l
ABCDEF a = AB, b = a b (1) AC (3) CD (2) AD (4) CE AF B C a A D b F E (1) AC = AB + BC = AB + AO = AB + ( AB + AF) = a + ( a + b) = 2 a + b (2) AD = 2 AO = 2( AB + AF) = 2( a + b) (3) CD = AF = b (4) CE
More informationexample2_time.eps
Google (20/08/2 ) ( ) Random Walk & Google Page Rank Agora on Aug. 20 / 67 Introduction ( ) Random Walk & Google Page Rank Agora on Aug. 20 2 / 67 Introduction Google ( ) Random Walk & Google Page Rank
More informationx V x x V x, x V x = x + = x +(x+x )=(x +x)+x = +x = x x = x x = x =x =(+)x =x +x = x +x x = x ( )x = x =x =(+( ))x =x +( )x = x +( )x ( )x = x x x R
V (I) () (4) (II) () (4) V K vector space V vector K scalor K C K R (I) x, y V x + y V () (x + y)+z = x +(y + z) (2) x + y = y + x (3) V x V x + = x (4) x V x + x = x V x x (II) x V, α K αx V () (α + β)x
More informationall.dvi
29 4 Green-Lagrange,,.,,,,,,.,,,,,,,,,, E, σ, ε σ = Eε,,.. 4.1? l, l 1 (l 1 l) ε ε = l 1 l l (4.1) F l l 1 F 30 4 Green-Lagrange Δz Δδ γ = Δδ (4.2) Δz π/2 φ γ = π 2 φ (4.3) γ tan γ γ,sin γ γ ( π ) γ tan
More informationA(6, 13) B(1, 1) 65 y C 2 A(2, 1) B( 3, 2) C 66 x + 2y 1 = 0 2 A(1, 1) B(3, 0) P 67 3 A(3, 3) B(1, 2) C(4, 0) (1) ABC G (2) 3 A B C P 6
1 1 1.1 64 A6, 1) B1, 1) 65 C A, 1) B, ) C 66 + 1 = 0 A1, 1) B, 0) P 67 A, ) B1, ) C4, 0) 1) ABC G ) A B C P 64 A 1, 1) B, ) AB AB = 1) + 1) A 1, 1) 1 B, ) 1 65 66 65 C0, k) 66 1 p, p) 1 1 A B AB A 67
More informationii
ii iii 1 1 1.1..................................... 1 1.2................................... 3 1.3........................... 4 2 9 2.1.................................. 9 2.2...............................
More information1 8, : 8.1 1, 2 z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = n i=1 a ii x 2 i + i<j 2a ij x i x j = ( x, A x), f =
1 8, : 8.1 1, z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = a ii x i + i
More information( )
7..-8..8.......................................................................... 4.................................... 3...................................... 3..3.................................. 4.3....................................
More informationDVIOUT-HYOU
() P. () AB () AB ³ ³, BA, BA ³ ³ P. A B B A IA (B B)A B (BA) B A ³, A ³ ³ B ³ ³ x z ³ A AA w ³ AA ³ x z ³ x + z +w ³ w x + z +w ½ x + ½ z +w x + z +w x,,z,w ³ A ³ AA I x,, z, w ³ A ³ ³ + + A ³ A A P.
More informationO E ( ) A a A A(a) O ( ) (1) O O () 467
1 1.0 16 1 ( 1 1 ) 1 466 1.1 1.1.1 4 O E ( ) A a A A(a) O ( ) (1) O O () 467 ( ) A(a) O A 0 a x ( ) A(3), B( ), C 1, D( 5) DB C A x 5 4 3 1 0 1 3 4 5 16 A(1), B( 3) A(a) B(b) d ( ) A(a) B(b) d AB d = d(a,
More informationn ξ n,i, i = 1,, n S n ξ n,i n 0 R 1,.. σ 1 σ i .10.14.15 0 1 0 1 1 3.14 3.18 3.19 3.14 3.14,. ii 1 1 1.1..................................... 1 1............................... 3 1.3.........................
More information16 B
16 B (1) 3 (2) (3) 5 ( ) 3 : 2 3 : 3 : () 3 19 ( ) 2 ax 2 + bx + c = 0 (a 0) x = b ± b 2 4ac 2a 3, 4 5 1824 5 Contents 1. 1 2. 7 3. 13 4. 18 5. 22 6. 25 7. 27 8. 31 9. 37 10. 46 11. 50 12. 56 i 1 1. 1.1..
More information1 n A a 11 a 1n A =.. a m1 a mn Ax = λx (1) x n λ (eigenvalue problem) x = 0 ( x 0 ) λ A ( ) λ Ax = λx x Ax = λx y T A = λy T x Ax = λx cx ( 1) 1.1 Th
1 n A a 11 a 1n A = a m1 a mn Ax = λx (1) x n λ (eigenvalue problem) x = ( x ) λ A ( ) λ Ax = λx x Ax = λx y T A = λy T x Ax = λx cx ( 1) 11 Th9-1 Ax = λx λe n A = λ a 11 a 12 a 1n a 21 λ a 22 a n1 a n2
More information(, Goo Ishikawa, Go-o Ishikawa) ( ) 1
(, Goo Ishikawa, Go-o Ishikawa) ( ) 1 ( ) ( ) ( ) G7( ) ( ) ( ) () ( ) BD = 1 DC CE EA AF FB 0 0 BD DC CE EA AF FB =1 ( ) 2 (geometry) ( ) ( ) 3 (?) (Topology) ( ) DNA ( ) 4 ( ) ( ) 5 ( ) H. 1 : 1+ 5 2
More informationA
A 2563 15 4 21 1 3 1.1................................................ 3 1.2............................................. 3 2 3 2.1......................................... 3 2.2............................................
More information18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C
8 ( ) 8 5 4 I II III A B C( ),,, 5 I II A B ( ),, I II A B (8 ) 6 8 I II III A B C(8 ) n ( + x) n () n C + n C + + n C n = 7 n () 7 9 C : y = x x A(, 6) () A C () C P AP Q () () () 4 A(,, ) B(,, ) C(,,
More information弾性定数の対称性について
() by T. oyama () ij C ij = () () C, C, C () ij ji ij ijlk ij ij () C C C C C C * C C C C C * * C C C C = * * * C C C * * * * C C * * * * * C () * P (,, ) P (,, ) lij = () P (,, ) P(,, ) (,, ) P (, 00,
More information() n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc n (
3 n nc k+ k + 3 () n C r n C n r nc r C r + C r ( r n ) () n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (4) n C n n C + n C + n C + + n C n (5) k k n C k n C k (6) n C + nc
More information0 (18) /12/13 (19) n Z (n Z ) 5 30 (5 30 ) (mod 5) (20) ( ) (12, 8) = 4
0 http://homepage3.nifty.com/yakuikei (18) 1 99 3 2014/12/13 (19) 1 100 3 n Z (n Z ) 5 30 (5 30 ) 37 22 (mod 5) (20) 201 300 3 (37 22 5 ) (12, 8) = 4 (21) 16! 2 (12 8 4) (22) (3 n )! 3 (23) 100! 0 1 (1)
More information4.6: 3 sin 5 sin θ θ t θ 2t θ 4t : sin ωt ω sin θ θ ωt sin ωt 1 ω ω [rad/sec] 1 [sec] ω[rad] [rad/sec] 5.3 ω [rad/sec] 5.7: 2t 4t sin 2t sin 4t
1 1.1 sin 2π [rad] 3 ft 3 sin 2t π 4 3.1 2 1.1: sin θ 2.2 sin θ ft t t [sec] t sin 2t π 4 [rad] sin 3.1 3 sin θ θ t θ 2t π 4 3.2 3.1 3.4 3.4: 2.2: sin θ θ θ [rad] 2.3 0 [rad] 4 sin θ sin 2t π 4 sin 1 1
More informationy π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a =
[ ] 9 IC. dx = 3x 4y dt dy dt = x y u xt = expλt u yt λ u u t = u u u + u = xt yt 6 3. u = x, y, z = x + y + z u u 9 s9 grad u ux, y, z = c c : grad u = u x i + u y j + u k i, j, k z x, y, z grad u v =
More information1 12 ( )150 ( ( ) ) x M x 0 1 M 2 5x 2 + 4x + 3 x 2 1 M x M 2 1 M x (x + 1) 2 (1) x 2 + x + 1 M (2) 1 3 M (3) x 4 +
( )5 ( ( ) ) 4 6 7 9 M M 5 + 4 + M + M M + ( + ) () + + M () M () 4 + + M a b y = a + b a > () a b () y V a () V a b V n f() = n k= k k () < f() = log( ) t dt log () n+ (i) dt t (n + ) (ii) < t dt n+ n
More informationORIGINAL TEXT I II A B 1 4 13 21 27 44 54 64 84 98 113 126 138 146 165 175 181 188 198 213 225 234 244 261 268 273 2 281 I II A B 292 3 I II A B c 1 1 (1) x 2 + 4xy + 4y 2 x 2y 2 (2) 8x 2 + 16xy + 6y 2
More informationA11 (1993,1994) 29 A12 (1994) 29 A13 Trefethen and Bau Numerical Linear Algebra (1997) 29 A14 (1999) 30 A15 (2003) 30 A16 (2004) 30 A17 (2007) 30 A18
2013 8 29y, 2016 10 29 1 2 2 Jordan 3 21 3 3 Jordan (1) 3 31 Jordan 4 32 Jordan 4 33 Jordan 6 34 Jordan 8 35 9 4 Jordan (2) 10 41 x 11 42 x 12 43 16 44 19 441 19 442 20 443 25 45 25 5 Jordan 26 A 26 A1
More information4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.
A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c
More informationR R 16 ( 3 )
(017 ) 9 4 7 ( ) ( 3 ) ( 010 ) 1 (P3) 1 11 (P4) 1 1 (P4) 1 (P15) 1 (P16) (P0) 3 (P18) 3 4 (P3) 4 3 4 31 1 5 3 5 4 6 5 9 51 9 5 9 6 9 61 9 6 α β 9 63 û 11 64 R 1 65 13 66 14 7 14 71 15 7 R R 16 http://wwwecoosaka-uacjp/~tazak/class/017
More information<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C60202E646F63>
電気電子数学入門 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/073471 このサンプルページの内容は, 初版 1 刷発行当時のものです. i 14 (tool) [ ] IT ( ) PC (EXCEL) HP() 1 1 4 15 3 010 9 ii 1... 1 1.1 1 1.
More informationD 24 D D D
5 Paper I.R. 2001 5 Paper HP Paper 5 3 5.1................................................... 3 5.2.................................................... 4 5.3.......................................... 6
More information.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =,
[ ] IC. r, θ r, θ π, y y = 3 3 = r cos θ r sin θ D D = {, y ; y }, y D r, θ ep y yddy D D 9 s96. d y dt + 3dy + y = cos t dt t = y = e π + e π +. t = π y =.9 s6.3 d y d + dy d + y = y =, dy d = 3 a, b
More information,2,4
2005 12 2006 1,2,4 iii 1 Hilbert 14 1 1.............................................. 1 2............................................... 2 3............................................... 3 4.............................................
More informationOABC OA OC 4, OB, AOB BOC COA 60 OA a OB b OC c () AB AC () ABC D OD ABC OD OA + p AB + q AC p q () OABC 4 f(x) + x ( ), () y f(x) P l 4 () y f(x) l P
4 ( ) ( ) ( ) ( ) 4 5 5 II III A B (0 ) 4, 6, 7 II III A B (0 ) ( ),, 6, 8, 9 II III A B (0 ) ( [ ] ) 5, 0, II A B (90 ) log x x () (a) y x + x (b) y sin (x + ) () (a) (b) (c) (d) 0 e π 0 x x x + dx e
More informationII R n k +1 v 0,, v k k v 1 v 0,, v k v v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ k σ dimσ = k 1.3. k
II 231017 1 1.1. R n k +1 v 0,, v k k v 1 v 0,, v k v 0 1.2. v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ kσ dimσ = k 1.3. k σ {v 0,...,v k } {v i0,...,v il } l σ τ < τ τ σ 1.4.
More information25 7 18 1 1 1.1 v.s............................. 1 1.1.1.................................. 1 1.1.2................................. 1 1.1.3.................................. 3 1.2................... 3
More information1/1 lim f(x, y) (x,y) (a,b) ( ) ( ) lim limf(x, y) lim lim f(x, y) x a y b y b x a ( ) ( ) xy x lim lim lim lim x y x y x + y y x x + y x x lim x x 1
1/5 ( ) Taylor ( 7.1) (x, y) f(x, y) f(x, y) x + y, xy, e x y,... 1 R {(x, y) x, y R} f(x, y) x y,xy e y log x,... R {(x, y, z) (x, y),z f(x, y)} R 3 z 1 (x + y ) z ax + by + c x 1 z ax + by + c y x +
More information17 ( ) II III A B C(100 ) 1, 2, 6, 7 II A B (100 ) 2, 5, 6 II A B (80 ) 8 10 I II III A B C(80 ) 1 a 1 = 1 2 a n+1 = a n + 2n + 1 (n = 1,
17 ( ) 17 5 1 4 II III A B C(1 ) 1,, 6, 7 II A B (1 ), 5, 6 II A B (8 ) 8 1 I II III A B C(8 ) 1 a 1 1 a n+1 a n + n + 1 (n 1,,, ) {a n+1 n } (1) a 4 () a n OA OB AOB 6 OAB AB : 1 P OB Q OP AQ R (1) PQ
More information2016 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 16 2 1 () X O 3 (O1) X O, O (O2) O O (O3) O O O X (X, O) O X X (O1), (O2), (O3) (O2) (O3) n (O2) U 1,..., U n O U k O k=1 (O3) U λ O( λ Λ) λ Λ U λ O 0 X 0 (O2) n =
More information50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq
49 2 I II 2.1 3 e e = 1.602 10 19 A s (2.1 50 2 I SI MKSA 2.1.1 r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = 3 10 8 m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq F = k r
More informationx () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x
[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),
More information2019 1 5 0 3 1 4 1.1.................... 4 1.1.1......................... 4 1.1.2........................ 5 1.1.3................... 5 1.1.4........................ 6 1.1.5......................... 6 1.2..........................
More information案内最終.indd
1 2 3 4 5 6 IC IC R22 IC IC http://www.gifu-u.ac.jp/view.rbz?cd=393 JR JR JR JR JR 7 / JR IC km IC km IC IC km 8 F HPhttp://www.made.gifu-u.ac.jp/~vlbi/index.html 9 Q01a N01a X01a K01a S01a T01a Q02a N02a
More informationIII 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F
III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F F 1 F 2 F, (3) F λ F λ F λ F. 3., A λ λ A λ. B λ λ
More information漸化式のすべてのパターンを解説しましたー高校数学の達人・河見賢司のサイト
https://www.hmg-gen.com/tuusin.html https://www.hmg-gen.com/tuusin1.html 1 2 OK 3 4 {a n } (1) a 1 = 1, a n+1 a n = 2 (2) a 1 = 3, a n+1 a n = 2n a n a n+1 a n = ( ) a n+1 a n = ( ) a n+1 a n {a n } 1,
More informationX G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2
More information1: *2 W, L 2 1 (WWL) 4 5 (WWL) W (WWL) L W (WWL) L L 1 2, 1 4, , 1 4 (cf. [4]) 2: 2 3 * , , = , 1
I, A 25 8 24 1 1.1 ( 3 ) 3 9 10 3 9 : (1,2,6), (1,3,5), (1,4,4), (2,2,5), (2,3,4), (3,3,3) 10 : (1,3,6), (1,4,5), (2,2,6), (2,3,5), (2,4,4), (3,3,4) 6 3 9 10 3 9 : 6 3 + 3 2 + 1 = 25 25 10 : 6 3 + 3 3
More information1 4 1 ( ) ( ) ( ) ( ) () 1 4 2
7 1995, 2017 7 21 1 2 2 3 3 4 4 6 (1).................................... 6 (2)..................................... 6 (3) t................. 9 5 11 (1)......................................... 11 (2)
More information() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
More informationSO(2)
TOP URL http://amonphys.web.fc2.com/ 1 12 3 12.1.................................. 3 12.2.......................... 4 12.3............................. 5 12.4 SO(2).................................. 6
More information(u(x)v(x)) = u (x)v(x) + u(x)v (x) ( ) u(x) = u (x)v(x) u(x)v (x) v(x) v(x) 2 y = g(t), t = f(x) y = g(f(x)) dy dx dy dx = dy dt dt dx., y, f, g y = f (g(x))g (x). ( (f(g(x)). ). [ ] y = e ax+b (a, b )
More informationi
i 3 4 4 7 5 6 3 ( ).. () 3 () (3) (4) /. 3. 4/3 7. /e 8. a > a, a = /, > a >. () a >, a =, > a > () a > b, a = b, a < b. c c n a n + b n + c n 3c n..... () /3 () + (3) / (4) /4 (5) m > n, a b >, m > n,
More information高校生の就職への数学II
II O Tped b L A TEX ε . II. 3. 4. 5. http://www.ocn.ne.jp/ oboetene/plan/ 7 9 i .......................................................................................... 3..3...............................
More information122 6 A 0 (p 0 q 0 ). ( p 0 = p cos ; q sin + p 0 (6.1) q 0 = p sin + q cos + q 0,, 2 Ox, O 1 x 1., q ;q ( p 0 = p cos + q sin + p 0 (6.2) q 0 = p sin
121 6,.,,,,,,. 2, 1. 6.1,.., M, A(2 R).,. 49.. Oxy ( ' ' ), f Oxy, O 1 x 1 y 1 ( ' ' ). A (p q), A 0 (p q). y q A q q 0 y 1 q A O 1 p x 1 O p p 0 p x 6.1: ( ), 6.1, 122 6 A 0 (p 0 q 0 ). ( p 0 = p cos
More information(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y
[ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)
More informationII Time-stamp: <05/09/30 17:14:06 waki> ii
II waki@cc.hirosaki-u.ac.jp 18 1 30 II Time-stamp: ii 1 1 1.1.................................................. 1 1.2................................................... 3 1.3..................................................
More informationnewmain.dvi
数論 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/008142 このサンプルページの内容は, 第 2 版 1 刷発行当時のものです. Daniel DUVERNEY: THÉORIE DES NOMBRES c Dunod, Paris, 1998, This book is published
More informationdynamics-solution2.dvi
1 1. (1) a + b = i +3i + k () a b =5i 5j +3k (3) a b =1 (4) a b = 7i j +1k. a = 14 l =/ 14, m=1/ 14, n=3/ 14 3. 4. 5. df (t) d [a(t)e(t)] =ti +9t j +4k, = d a(t) d[a(t)e(t)] e(t)+ da(t) d f (t) =i +18tj
More information1 29 ( ) I II III A B (120 ) 2 5 I II III A B (120 ) 1, 6 8 I II A B (120 ) 1, 6, 7 I II A B (100 ) 1 OAB A B OA = 2 OA OB = 3 OB A B 2 :
9 ( ) 9 5 I II III A B (0 ) 5 I II III A B (0 ), 6 8 I II A B (0 ), 6, 7 I II A B (00 ) OAB A B OA = OA OB = OB A B : P OP AB Q OA = a OB = b () OP a b () OP OQ () a = 5 b = OP AB OAB PAB a f(x) = (log
More informationL1-a.dvi
27 Q C [ ] cosθ sinθ. A θ < 2π sinθ cosθ A. A ϕ A, A cosϕ cosθ sinθ cosθ sinθ A sinθ cosθ sinθ +cosθ A, cosθ sinθ+sinθ+cosθ 2 + 2 cosθ A 2 A,A cosθ sinθ 2 +sinθ +cosθ 2 2 cos 2 θ+sin 2 θ+ 2 sin 2 θ +cos
More informatione a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1, σ,..., σ N ) i σ i i n S n n = 1,,
01 10 18 ( ) 1 6 6 1 8 8 1 6 1 0 0 0 0 1 Table 1: 10 0 8 180 1 1 1. ( : 60 60 ) : 1. 1 e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1,
More informationS K(S) = T K(T ) T S K n (1.1) n {}}{ n K n (1.1) 0 K 0 0 K Q p K Z/pZ L K (1) L K L K (2) K L L K [L : K] 1.1.
() 1.1.. 1. 1.1. (1) L K (i) 0 K 1 K (ii) x, y K x + y K, x y K (iii) x, y K xy K (iv) x K \ {0} x 1 K K L L K ( 0 L 1 L ) L K L/K (2) K M L M K L 1.1. C C 1.2. R K = {a + b 3 i a, b Q} Q( 2, 3) = Q( 2
More informationx x x 2, A 4 2 Ax.4 A A A A λ λ 4 λ 2 A λe λ λ2 5λ + 6 0,...λ 2, λ 2 3 E 0 E 0 p p Ap λp λ 2 p 4 2 p p 2 p { 4p 2 2p p + 2 p, p 2 λ {
K E N Z OU 2008 8. 4x 2x 2 2 2 x + x 2. x 2 2x 2, 2 2 d 2 x 2 2.2 2 3x 2... d 2 x 2 5 + 6x 0 2 2 d 2 x 2 + P t + P 2tx Qx x x, x 2 2 2 x 2 P 2 tx P tx 2 + Qx x, x 2. d x 4 2 x 2 x x 2.3 x x x 2, A 4 2
More information1 θ i (1) A B θ ( ) A = B = sin 3θ = sin θ (A B sin 2 θ) ( ) 1 2 π 3 < = θ < = 2 π 3 Ax Bx3 = 1 2 θ = π sin θ (2) a b c θ sin 5θ = sin θ f(sin 2 θ) 2
θ i ) AB θ ) A = B = sin θ = sin θ A B sin θ) ) < = θ < = Ax Bx = θ = sin θ ) abc θ sin 5θ = sin θ fsin θ) fx) = ax bx c ) cos 5 i sin 5 ) 5 ) αβ α iβ) 5 α 4 β α β β 5 ) a = b = c = ) fx) = 0 x x = x =
More informationii-03.dvi
2005 II 3 I 18, 19 1. A, B AB BA 0 1 0 0 0 0 (1) A = 0 0 1,B= 1 0 0 0 0 0 0 1 0 (2) A = 3 1 1 2 6 4 1 2 5,B= 12 11 12 22 46 46 12 23 34 5 25 2. 3 A AB = BA 3 B 2 0 1 A = 0 3 0 1 0 2 3. 2 A (1) A 2 = O,
More information2S III IV K A4 12:00-13:30 Cafe David 1 2 TA 1 appointment Cafe David K2-2S04-00 : C
2S III IV K200 : April 16, 2004 Version : 1.1 TA M2 TA 1 10 2 n 1 ɛ-δ 5 15 20 20 45 K2-2S04-00 : C 2S III IV K200 60 60 74 75 89 90 1 email 3 4 30 A4 12:00-13:30 Cafe David 1 2 TA 1 email appointment Cafe
More information13 0 1 1 4 11 4 12 5 13 6 2 10 21 10 22 14 3 20 31 20 32 25 33 28 4 31 41 32 42 34 43 38 5 41 51 41 52 43 53 54 6 57 61 57 62 60 70 0 Gauss a, b, c x, y f(x, y) = ax 2 + bxy + cy 2 = x y a b/2 b/2 c x
More information