1 II, II.,.,.,. 1 K R C, ( ) V, : ➀ u, v V u + v V. ➁ u V c K cu V., u, v, w V a, b K (1) (8). (1) u+v=v+u (2) (u+v)+w=u+(v+w) (3) o V, v + o =
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1 1 II, II,,, 1 K R C, ( ) 11 V, : ➀ u, v V u + v V ➁ u V c K cu V, u, v, w V a, b K (1) (8) (1) u+v=v+u (2) (u+v)+w=u+(v+w) (3) o V, v + o = v (4) a(bu) = (ab)u (5) (a + b)u = au + bu (6) a(u + v) = au + av (7) 1u = u (8) 0u = o, V K, V 1 (1) R n R 11 (1) (8), ➀,➁ ➀ u 1 u 2 v 1 v 2 u, v R n, u =, v = (u 1, u 2,, u n, v 1, v 2,, v n ) u n v n u 1 + v 1 u 2 + v 2, u + v =,, u + v R n u n + v n ➁ cu 1 cu 2 u 1 u 2 u R n c R ( 1), u = (u 1, u 2,, u n ) cu = u n cu n,, cu R n ( 1), R n R, K R
2 2 (2) R n C, 11 ➁ (1) c C (m, n) M mn (R) I C(I : R) C (3) C n C, R (4) n R n [x], (m, n) M mn (R), [a, b] C([a, b]; R) R (5) R 2 R 3,, R C R 2 R 3 (6) {o} ( ) R, C 12 V K, W V W u 1, u 2 W, c 1, c 2 K c 1 u 1 + c 2 u 2 W, W V K V W, V K V V, V W,, V 13 K V a 1, a 2,, a n c 1 a 1 +c 2 a 2 + +c n a n = o (, c 1, c 2,, c n K ) c 1 = c 2 = = c n = 0 a 1, a 2,, a n ( 2) a 1, a 2,, a n, a 1, a 2,, a n 14 a 1, a 2,, a r, b 1, b 2,, b r K n n r A = (a 1 a 2 a r ), B = (b 1 b 2 b r ) A B, (1), (2) (1) b r = c 1 b 1 + c 2 b c r 1 b r 1 (c 1, c 2,, c r 1 ), a r = c 1 a 1 + c 2 a c r 1 a r 1 (2) b 1, b 2,, b r, a 1, a 2,, a r 15 1 a 1, a 2,, a n K n n A = (a 1 a 2, (1), (2) a n ) (1) a 1, a 2,, a n (2) A a 1, a 2,, a r K n n r A = (a 1 a 2, (1), (2) a r ) (1) a 1, a 2,, a r (2) rank A = r 17 K V u 1, u 2,, u n ➀ u 1, u 2,, u n ➁ V v u 1, u 2,, u n, V v v = c 1 u 1 + c 2 u c n u n c 1, c 2,, c n K ( 2),,, : ➀ K V a 1, a 2,, a n c 1 a 1 + c 2 a c n a n = o c 1, c 2,, c n K c 1 = c 2 = = c n = 0 a 1, a 2,, a n ➁ K V a 1, a 2,, a n c 1 a 1 + c 2 a c n a n = o c 1, c 2,, c n K c 1 = c 2 = = c n = 0 a 1, a 2,, a n
3 3 2, {u 1, u 2,, u n } V 18 K V u 1, u 2,, u n, u 1, u 2,, u n = {c 1 u 1 + c 2 u c n u n c 1, c 2,, c n K}, u 1, u 2,, u n V V u 1, u 2,, u n u 1, u 2,, u n, ➁, V = u 1, u 2,, u n, 19 V K, V, 110 K V V, dim V, V dim V = n, V n, V = {o}, V V K, V n a 1, a 2,, a n r v 1, v 2,, v r, n > r, a 1, a 2,, a n v 1, v 2,, v r, a 1, a 2,, a n, 112 V K n {w 1, w 2,, w n } V, V n, (R n, C n, R n [x] ),, 113 W 1, W 2 K V, W 1 W 2 W 1 W W 1, W 2 K V, W 1 W 2 W 1 + W 2 = {w 1 + w 2 w 1 W 1, w 2 W 2 } K V W 1, W 2, W 1 W 2 W 1 + W 2 V (, ) 115 W 1, W 2 K V, dim W 1 + dim W 2 = dim(w 1 + W 2 ) dim(w 1 W 2 )
4 a A b B f, f A B, f : A B, b = f (a) 22 U, V K f : U V ➀ u 1, u 2 U f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ) ➁ u U, c K f (cu) = c f (u) 2, f ➀, ➁ : u 1, u 2 U, c 1, c 2 K, f (c 1 u 1 + c 2 u 2 ) = c 1 f (u 1 ) + c 2 f (u 2 ) 23 U, V K f : U V, f, f Im f = {v V v = f (u) u U }, ker f = {u U f (u) = o} 24 U, V K, f : U V, dim U = dim Im f + dim ker f
5 R V R x, y V (x, y), (1) (3), (x, y) x,y (1) ( ) x, y, z V, c R (x + y, z) = (x, z) + (y, z), (x, y + z) = (x, y) + (x, z), (cx, y) = c(x, y), (x, cy) = c(x, y) (2) ( ) x, y V, (x, y) = (y, x) (3) ( ) x V, (x, x) 0 x = o c = a + bi (a, b R), c = a bi c 31 C V C x, y V (x, y), (1) (3), (x, y) x,y (1) ( ) x, y, z V, c C ( 3) (x + y, z) = (x, z) + (y, z), (x, y + z) = (x, y) + (x, z), (cx, y) = c(x, y), (x, cy) = c(x, y) (2) ( ) x, y V, (x, y) = (y, x) (3) ( ) x V, (x, x) 0 x = o V, V 32 ( ) V, x V x x = (x, x) 33 V, x, y V, (x, y) x y 34 V, (1) x V, x 0 (2) x V c K, cx = c x (3) x, y V, x + y x + y 35 R V, x, y V, (x, y) = x y cos θ, 0 θ π θ x, y, V x, y (x, y) = 0, x y ( 3)
6 6 36 a 1, a 2,, a n R n, A = (a 1 a 2 a n ), G = t AA A, t A A ➀, a 1 2 (a 1, a 2 ) (a 1, a 2 ) (a 1, a n ) (a 2, a 1 ) a 2 2 (a 2, a 3 ) (a 2, a n ) t AA = (a 3, a 1 ) (a 3, a 2 ) a 3 2 (a 3, a n ) (a n, a 1 ) (a n, a 2 ) (a n, a 3 ) a n 2 ➁, t ( t AA) = t AA 37 t PP = E P 38 P = (p 1 p 2 p n ) 2 ➀ p 1, p 2,, p n 1, p 1 = p 2 = = p n = 1 ➁ p 1, p 2,, p n 2, i j i, j = 1, 2,, n, (p i, p j ) = 0 ➀, ➁, (p i, p j ) = δ ij (i, j = 1, 2,, n) 39 V n, p 1, p 2,, p n V 3, {p 1, p 2,, p n } V : ➀ p 1, p 2,, p n 1 ➁ p 1, p 2,, p n 2 ➂ {p 1, p 2,, p n } V 310 V a 1, a 2,, a n V, a 1, a 2,, a n ➀ P = (p 1 p 2 p n ) ➁ {p 1, p 2,, p n } R n
7 7 312 V, {a 1, a 2,, a n } V, b 1, b 2,, b n, {b 1, b 2,, b n} (I) b 1 = a 1, b 1 = b 1 b 1 j 1 (II) j = 2, 3,, n, b j = a j (a j, b i )b i, b j = b j b j, R 3 : i=1 313 R 3 {a 1, a 2, a 3 } R 3, b 1, b 2, b 3, {b 1, b 2, b 3 } (I) b 1 = a 1, b 1 = b 1 b 1 (II) b 2 = a 2 (a 2, b 1 )b 1, b 2 = b 2 b 2 (III) b 3 = a 3 (a 3, b 1 )b 1 (a 3, b 2 )b 2, b 3 = b 3 b n A, Ax = λx, x o λ x C n, λ A, x λ ➀ det(λe A) = 0, λ ➁ ➀, (λe A)x = o x (, x o ) A n, det(λe A) n, det(λe A) 42, ➀ λ 1, λ 2,, λ r, det(λe A) = (λ λ 1 ) n 1 (λ λ2 ) n2 (λ λ r ) n r, j = 1, 2,, r n j λ j ➁ A λ, V λ = {x C n Ax = λx} λ ( 4) V λ λ 43 λ 1, λ 2,, λ r A, x 1, x 2,, x r λ 1, λ 2,, λ r, x 1, x 2,, x r ( 4) V λ W(λ, A), V(λ)
8 n A, n P, λ λ P AP = (1), A ( 5) 52 n A, 2 ➀ A, P = (p 1 p 2 p n ), (1) ➁ A n p 1, p 2,, p n λ 1, λ 2,, λ n, p 1, p 2,, p n A, n A P = (p 1 p 2 p n ), : AP = A(p 1 p 2 p n ) = (Ap 1 Ap 2 Ap n ), (2) λ 1 λ 1 P λ n = (p 1 p 2 p n ) λ n = (λ 1p 1 λ n p n ) (3) ➀ ➁ (1), AP = P λ 1 λ n, (2), (3), (Ap 1 Ap n ) = (λ 1 p 1 λ n p n ), Ap 1 = λ 1 p 1,, Ap n = λ n p n, λ 1,, λ n A, p 1,, p n λ 1,, λ n p 1,, p n, P, det P 0 ➁ ➀ p 1,, p n A λ 1,, λ n n, Ap 1 = λ 1 p 1,, Ap n = λ n p n, (2), (3) λ 1, AP = P λ n P 1 ( 6) (1), A, A λ 1, λ 2,, λ n p 1, p 2,, p n, P = (p 1 p 2 p n ), (1) n A n λ 1, λ 2,, λ n, A, ( 5) ( 6) P 1, p 1,, p n, det P 0
9 9,, n A, det(λe A) = (λ λ 1 ) n 1 (λ λ2 ) n2 (λ λ r ) n r, λ 1, λ 2,, λ r, 2 ➀ A ➁ j = 1, 2,, r, dim V λj = n j, 55 λ 0 n A, n 0 λ 0, dim V λ0 n 0 dim V λ0 = d, V λ0 C n, d n C n {v 1, v 2,, v n } {v 1,, v d } ( d ) V λ0 P = (v 1 v 2 v n ), v 1, v 2,, v n P, v 1, v 2,, v d λ 0,, AP = (Av 1 Av d Av d+1 Av n ) = (λ 0 v 1 λ 0 v d Av d+1 Av n ) Av d+1,, Av n C n, v 1, v 2,, v n, j = d + 1,, n, Av j = b 1,j v 1 + b 2,j v b n,j v n b 1,j,, b n,j C, λ 0 b 1,d+1 b 1,n AP = (v 1 v 2 v n ) λ 0 = P λ 0 b 1,d+1 b 1,n λ 0 b n,d+1 b n,n b n,d+1 b n,n ( ) ( 0 ), P 1 λ0 E d B AP =, O C E d d, B d (n d), C (m d), det(λe A) = (λ λ 0 ) d det(λe n d C),, λ 0 d ( 7), n 0 d,, 1 2 ➁, 2 ➁, ➀ ➁ 52, A n, r j=1 dim V λj = n, A n, r j=1 n j = n ( ), r j=1 dim V λj = r j=1 n j 55, j = 1, 2,, r dim V λj n j, j <, ( ), j = 1, 2,, r dim V λj = n j ➁ ➀ j = 1, 2,, r dim V λj = n j, j = 1, 2,, r, V λj p j,1,, p j,nj (V λj = p j,1,, p j,nj ) n 1 + n n r = n, n 43 1, 52 A ( 7) det(λe n d C) (λ λ 0 )
10 10 54 ➁, ( 8) 56 n λ 1, λ 2,, λ n, P, P 1 AP ( 9) : λ 1 0 λ P 1 2 AP = n n = 1 ( ) n 1, n λ 1 p 1, P = (p 1 p 2 p n ) p 2,, p n, A P = (Ap 1 Ap 2 Ap n ) = (λ 1 p 1 Ap 2 Ap n ) P, {p 1, p 2,, p n } C n, Ap 2,, Ap n p 1,, p n, j = 2, 3,, n, Ap j = b 1j p 1 + b 2j p b nj p n b 1j, b 2j,, b nj K, λ 1 b 12 b 13 b 1n λ 1 b 12 b 13 b 1n 0 b 22 b 23 b 2n 0 b 22 b 23 b 2n A P = (p 1 p 2 p n ) = P 0 b n2 b n3 b nn 0 b n2 b n3 b nn ( ), P 1 λ1 A P = (B n 1 ), o B λe n A = P 1 λe n A P = λe n P 1 A P = λ λ 1 o λe n 1 B = (λ λ 1) λe n 1 B, B A λ 2, λ 3,, λ n B n 1,, Q, λ 2 0 λ Q 1 3 BQ = ( ) 1 t o R =, P = PR, o Q ( ) ( ) ( 1 P 1 t o 1 AP = P 1 o Q 1 A P t o 1 t o = o Q o Q 1 ) ( ) ( ) ( λ1 1 t o λ1 = o B o Q o Q 1 BQ ) ( 8), ( 9),,
11 11, P 1 AP = λ 1 0 λ 2 57 A n, A λ 1, λ 2,, λ n λ 1, λ 2,, λ n, P, t PAP : λ 1 0 λ t 2 PAP =,, 56, P, {p 1, p 2,, p n } ( 10) 6 61 t A = A A 62 n A, ➀ A ➁ A ➂ λ 1, λ 2,, λ n A, λ 1, λ 2,, λ n p 1, p 2,, p n {p 1, p 2,, p n }, P = (p 1 p 2 p n ), λ λ t PAP =, (4) ( 10),
2 (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
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
1 2 3 1 2 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 % 23 24 25 26 27 28 29 30 31 32 33 34 35 4 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68
1 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
h16ﾏｽﾀｰｾﾝﾀｰ報告書（神奈川県支部）
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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
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1W II K =25 A (1) office(a439) (2) A4 etc. 12:00-13:30 Cafe David 1 2 TA appointment Cafe D
1W II K200 : October 6, 2004 Version : 1.2, kawahira@math.nagoa-u.ac.jp, http://www.math.nagoa-u.ac.jp/~kawahira/courses.htm TA M1, m0418c@math.nagoa-u.ac.jp TA Talor Jacobian 4 45 25 30 20 K2-1W04-00
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数学Ⅱ演習（足助・09夏）
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() 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 ()
(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
.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
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(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
[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s
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[ ]. lim e 3 IC ) s49). y = e + ) ) y = / + ).3 d 4 ) e sin d 3) sin d ) s49) s493).4 z = y z z y s494).5 + y = 4 =.6 s495) dy = 3e ) d dy d = y s496).7 lim ) lim e s49).8 y = e sin ) y = sin e 3) y =
x 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
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2 T140073 1 2 ax 2 + 2bxy + cy 2 + dx + ey + f = 0 a + b + c > 0 a, b, c A xy u = u 0 + a cos θ, v = v 0 + b sin θ 0 θ 2π u = u 0 ± a cos θ, v = v 0 + b tan θ π 2 < θ < π 2 u = u 0 + 2pt, v = v 0 + pt
31 4 MATLAB A, B R 3 3 A = , B = mat_a, mat_b >> mat_a = [-1, -2, -3; -4, -5, -6; -7, -8, -9] mat_a =
![31 4 MATLAB A, B R 3 3 A = , B = mat_a, mat_b >> mat_a = [-1, -2, -3; -4, -5, -6; -7, -8, -9] mat_a =](/thumbs/99/140476730.jpg "31 4 MATLAB A, B R 3 3 A = , B = mat_a, mat_b >> mat_a = [-1, -2, -3; -4, -5, -6; -7, -8, -9] mat_a =")
3 4 MATLAB 3 4. A, B R 3 3 2 3 4 5 6 7 8 9, B = mat_a, mat_b >> mat_a = [-, -2, -3; -4, -5, -6; -7, -8, -9] 9 8 7 6 5 4 3 2 mat_a = - -2-3 -4-5 -6-7 -8-9 >> mat_b = [-9, -8, -7; -6, -5, -4; -3, -2, -]
1. (8) (1) (x + y) + (x + y) = 0 () (x + y ) 5xy = 0 (3) (x y + 3y 3 ) (x 3 + xy ) = 0 (4) x tan y x y + x = 0 (5) x = y + x + y (6) = x + y 1 x y 3 (
1 1.1 (1) (1 + x) + (1 + y) = 0 () x + y = 0 (3) xy = x (4) x(y + 3) + y(y + 3) = 0 (5) (a + y ) = x ax a (6) x y 1 + y x 1 = 0 (7) cos x + sin x cos y = 0 (8) = tan y tan x (9) = (y 1) tan x (10) (1 +
a n a n ( ) (1) a m a n = a m+n (2) (a m ) n = a mn (3) (ab) n = a n b n (4) a m a n = a m n ( m > n ) m n 4 ( ) 552
3 3.0 a n a n ( ) () a m a n = a m+n () (a m ) n = a mn (3) (ab) n = a n b n (4) a m a n = a m n ( m > n ) m n 4 ( ) 55 3. (n ) a n n a n a n 3 4 = 8 8 3 ( 3) 4 = 8 3 8 ( ) ( ) 3 = 8 8 ( ) 3 n n 4 n n